I am a student learning the Russian language and interested in the Russian history and culture, and I have read many Russian articles, blogs, and materials about Russia, including discussions on Russian forums. I will now write my impressions and observations relevant to your question, and I hope that Russian users of SE will correct all inaccuracies in my answer if there are any.
The short answer to your question is that Russians do not care much about Putin's rhetoric about Western multiculturalism.
Let me make a few points and explain them in order to provide a detailed answer.
1. Russians tend to be considerably less interested in politics than people in the West are, owing to peculiarities of the political landscape in Russia.
Churchill said long time ago that Kremlin intrigues are comparable to a bulldog fight under a rug: an observer only hears the growling, and when he sees the bones fly out from beneath it is obvious who won. This aphorism seems to be still applicable to Russia's political life. Many Russians see their government and the ruling elite as a mafia that maintains order and stability on the territory it controls. In view of many Russians, Russia's real political life mainly occurs inside that mafia, behind closed doors, and is thus hidden from ordinary people, who only see which person gets which position (minister, mayor, etc.).
Russian newspapers, news agencies, and TV channels refrain from seriously criticizing the government and supporting the so-called opposition.
The activity of the opposition seems pretty pointless as the opposition does not have any real power and, in particular, cannot do anything when cheated against in elections and courts. Elections in Russia are merely an imitation of democracy: votes are manipulated, and politicians disliked by the government are often not allowed to participate in elections as candidates. Russian courts are not independent from the government and obey when given orders "from above."
Influenced or brainwashed by mass media, many Russians believe that the boss, Putin, knows what he is doing, and that an alternative leader might lead the country to chaos, so they vote for Putin and the ruling party, seeing no alternative and remembering the chaos of the 1990s.
With the median salary in Russia being 500$ per month or so, many Russians barely make the ends meet and have neither energy nor motivation to think and learn much about politics. [In response to a comment below, I will explain here the cost of living in Russia. Prices for food, clothes and electronics in Russia are roughly the same as in Western countries such as Germany and France (if we compare products of the same quality), whilst renting an accommodation in Russia is considerably cheaper than in Western Europe, but takes a considerable portion of the median salary in any given Russian city. The so-called прожиточный минимум (cost of minimal consumption for an adult), which is an officially defined and legally used quantity that includes food, clothes, and bills, but does not include renting an accommodation, is officially about 150$ per month (with some variations across the country), but is far from being enough to maintain a healthy lifestyle of a middle-aged adult.]
2. Practically no one in Russia expects Putin's policy about ethnic groups to change, as that policy has always been very consistent.
The policy has always been to ensure stability in Russia and, in particular, extirpate all extremist, radical, and separatist movements as well as prevent conflicts between ethnic groups. In the Russian criminal law, there is a special section for "igniting conflicts between ethnic groups."
Since the collapse of the Soviet Union there have been no visible attempts by the Russian government to restrict the freedom of religious beliefs. In many Russian cities, including Moscow, you can find churches, mosques, and synagogues, all peacefully co-existing together.
Likewise, there have been no visible attempts by the government to suppress cultures or languages of ethnic groups. Languages of various ethnic groups are officially recognized, have their own alphabets, and are freely used. Examples are the Tatar, Chuckchi, and Kalmyk languages, which have their own alphabets somewhat different from the Russian alphabet. The only "suppression" that comes to mind is that the government did not let the Tatars officially adopt a Latin-based aplhabet, so they use a Cyrillic-based one instead. The Russian language is the lingua franca in Russia, so very few people living in Russia do not speak Russian, but many Russian citizens are bilingual, and the Russian government does not have any problem with it.
Putin is okay with existence of regions with a somewhat different culture as long as they do not cause trouble. Governments of such regions demonstrate loyalty to Putin and do their best not to cause trouble. A number of such regions (e.g., Chechnya and Chukotka) are financially helped by the Russian government, getting more money from the federal budget than giving back, and thus have no reason to complain about the treatment by the Russian government.
Concerning specifically Russian Muslims, here is how Putin himself expressed his attitude towards them:
Today, traditional Islam is an integral part of Russia's spiritual life. Islam's humanist values, like the values of our other traditional religions, teach people compassion, justice and care for our loved ones. We place great value on these things. ... It is important to educate Muslim youth in traditional Islamic values and prevent attempts to impose on us world outlooks that are alien to us and have nothing to do with genuine Islam. Let me say that the authorities will continue to assist in reviving Russia's system of Islamic theological schools and religious education. ... Of course, we must continue expanding the network of Muslim cultural and educational centres. Their aim is to bring Muslims together, impart to them the spiritual, cultural and moral code inherent to traditional Islam in Russia, help to resolve common problems, and take part in youth education. I note the big role that Muslims and above all their spiritual leaders play in strengthening interethnic and interfaith harmony. Their rejection and condemnation of all forms of fundamentalism and radicalism have made a major contribution to the fight against nationalism and religious extremism. (Source)
In short, Putin views Russian Muslims positively as long as they help him maintain order and stability and do not cause trouble.
There are no indications of any upcoming change, so Russians do not really care what Putin rhetorically says to the West about Western multiculturalism.
3. Many Russians do not seriously take Putin's rhetoric speeches.
Many Russians understand that his speeches do not necessarily convey what he really thinks or is up to. For example, in the very same interview in which he made his remark about multiculturalism he also said the following:
You know, first of all, we do not have oligarchs anymore. Oligarchs are those who use their proximity to the authorities to receive super profits. We have large companies, private ones, or with government participation. But I do not know of any large companies that get preferential treatment from being close to the authorities, these are practically non-existent.
Many Russians laughed reading this, as I see on the Internet. To them, nothing can be further from the truth. (See examples of reaction here.)
4. Putin's remark about multiculturalism was made in relation to migration, and his actual message seems to be merely that a massive inflow of troublesome migrants to a country can cause trouble in that country.
Let us read the exact words of Putin about multiculturalism in that interview:
Vladimir Putin: ... There is also the so-called liberal idea, which has outlived its purpose. Our Western partners have admitted that some elements of the liberal idea, such as multiculturalism, are no longer tenable. When the migration problem came to a head, many people admitted that the policy of multiculturalism is not effective and that the interests of the core population should be considered. Although those who have run into difficulties because of political problems in their home countries need our assistance as well. That is great, but what about the interests of their own population when the number of migrants heading to Western Europe is not just a handful of people but thousands or hundreds of thousands?
Lionel Barber: Did Angela Merkel make a mistake?
Vladimir Putin: Cardinal mistake. One can criticise Trump for his intention to build a wall between Mexico and the United States. It could be going too far. Yes, maybe so. I am not arguing about this point. But he had to do something about the huge inflow of migrants and narcotics. Nobody is doing anything. They say this is bad and that is bad as well. Tell me, what is good then? What should be done? Nobody has proposed anything. I do not mean that a wall must be built or tariffs raised by 5 percent annually in the economic relations with Mexico. This is not what I am saying, yet something must be done. He is at least looking for a solution. What am I driving at? Those who are concerned about this, ordinary Americans, they look at this and say, Good for him, at least he is doing something, suggesting ideas and looking for a solution. As for the liberal idea, its proponents are not doing anything. They say that all is well, that everything is as it should be. But is it? They are sitting in their cosy offices, while those who are facing the problem every day in Texas or Florida are not happy, they will soon have problems of their own. Does anyone think about them? The same is happening in Europe. I discussed this with many of my colleagues, but nobody has the answer. The say they cannot pursue a hard-line policy for various reasons. Why exactly? Just because. We have the law, they say. Well, then change the law! We have quite a few problems of our own in this sphere as well. We have open borders with the former Soviet republics, but their people at least speak Russian. Do you see what I mean? And besides, we in Russia have taken steps to streamline the situation in this sphere. We are now working in the countries from which the migrants come, teaching Russian at their schools, and we are also working with them here. We have toughened the legislation to show that migrants must respect the laws, customs and culture of the country.
As you see, Putin talks about migrants, not ethnic minorities integrated into a country's life, and it is hard to see any implications about possible future measures against ethnic minorities that have always been living in Russia and have learned not to cause trouble. Putin seems to have chosen the word "multiculturalism" rather carelessly, and his message seems to be merely that a massive inflow of troublesome migrants to a country can cause trouble in that country.
I performed a search in Google to see how Russians had reacted to Putin's words about multiculturalism, and here is a typical example of reaction. The author, a popular blogger, sees hypocrisy in Putin's words as Russia has about two million illegal immigrants and open borders with poor neighboring countries, whose citizens do not need a visa to travel to Russia.
To conclude, my impression is that Russians do not draw any serious conclusions from Putin's words about Western multiculturalism. They see those words rather as rhetoric of little significance for them personally.
UPDATE: In response to the OP's comment below, I am adding citations of articles in Russian media in order to evidence the points made by me.
Widespread political apathy: link1, link2
"Mafia state": link3, link4, link5, link6
Manipulations in elections: link7, link8
Denials to participate in elections: link9, link10
Economic hardship of ordinary Russians: link11, link12, link13
Government policy on ethnic groups: link14, link15
Russians do not seriously take Putin's rhetoric: link16, link17, link18