There are multiple aspects to consider. There is also an additional question embedded into your question regarding why European governments are playing a role in an alliance to counter China militarily. This answer will also address that question, but it really comes down to competition on multiple levels.
The ideological differences between the Western governments and the Chinese governments are well known, but this has low relevance explaining why European governments are interested this region. In comparison to Cold War 1, the Chinese government does not seek to export socialism as a form of government to other countries. But, there is some interest among the western governments in getting Indo Pacific countries to adopt western like values on issues such as human rights, intellectual property, democracy, etc.
China differs in that it generally does not attempt to encourage countries to adopt domestic reforms in return for economic cooperation. All developing economies want trade agreements on the best possible terms with developed economies. This gives western countries the ability to offer incentives to developing countries for political reforms and influence in return for trade agreements.
It does not work like this today because China itself is still an emerging country, but you can imagine a day where a developing country will have the option to align itself with one of the 3 major spheres (the US, Europe, or China). By introducing another competitor into the picture, a government of a developing country may then have the option to have a major trading partner without having any foreign influence on its domestic political structure. Hence, it's not quite the same as the "domino effect" during Cold War 1, but in general, western governments would feel more comfortable if they were not the only liberal democracies surrounded by a world dominated by authoritarian or socialist countries.
As you mention, the main thrust of the tension is between the US and China. However, the US is the primary component of NATO, and the US doesn't have as much strategic interest in Europe as it does in the Western Pacific. This creates a problem for European governments that heavily rely on the US for defense.
Meanwhile, the US does not like having a public perception that it is a superpower in a uni-polar world. For example, the Iraq War was almost entirely an American conflict (although the UK also supported it), but the US tried and failed to get NATO on its side. Instead, George W Bush offered several Baltic countries support for entering NATO in return for joining the American side even though their military contribution was very minor due to their size. Why? Because the US didn't want the war to be a US vs Iraq war. It wanted a "Global Coalition" vs Iraq war.
Hence, something similar is happening today where the US wants European support for both international and domestic perception of its actions regarding China even though the ability of the European navies to operate meaningfully so far away from home is quite limited today. America's actions in the South China Sea are much more palatable internationally and domestically if it seen as part of an international coalition rather than a lone superpower. In return, the US is going to remain steadfast in its NATO role even though its interests in Europe are much lower than in the past.
The European interests in the Indo Pacific are not much different than their interests in Africa. Like other governments, the EU's primary mission is to preserve (and even improve) the economic way of life of its workers in the agriculture, manufacturing, and services sectors. Although less relevant for the US which has a large resource endowment, the governments of the major developed economies need access to raw materials and want to export services and manufactured goods on the best possible asymmetric terms for their country that they can negotiate.
Both parties might benefit from trade agreements in theory, but in practice, one party benefits more than the other. If I'm the economically stronger power and I'm your only option, I can force you to open your market to my cars produced with subsidized electricity in return for me opening my market to your raw aluminum. We both gain, but you will never have the opportunity to build up a domestic car industry. So, my economy keeps the most valuable part of the value chain, and your economy is based purely on job lite resource extraction.
I might also be interested in low cost labor (Indo Pacific) as well in the place of aluminum above, but the principle is the same; I want to keep the most valuable parts of the value chain at home domestically and get market access for my high value exports.
However, suppose a third country enters the picture and offers you more favorable terms such as investment and a promise his/her state owned company will make some cars in your country domestically. I can compete with that country, but I will never get as good a deal as I have today. In some cases, an entire industry may shut down in my country if you make a trade deal with another economy, and I'm not able to get the raw commodities my economy needs. We're not exactly at that point yet, but you can see why the EU views China as a "systemic rival".
Of the three aspects above, economics has the most explanatory power in the events happening today. On the topic of trade with China, that model worked really well when China was providing low cost, mass labor on the lower parts of the value chain. The problem is that now China is moving into high value parts of the value chain, which means its skilled work force could (in theory) be in direct competition with the German and French skilled work force, and that is a big problem. This is also why you see a new hysteria over semiconductor, AI, etc. protectionism. But if you look at each aspect of the situation above, there are basically two global teams forming, and Europe's interests are much more aligned with the non-China team on every single aspect.