Cultural and legal aspects aside, some interesting factoids can be gleaned by looking at the EU-Turkey trade, especially broken down by EU country. Turkey is a large-ish trading partner of the Eastern EU countries. Bulgaria tops the list percentage-wise with a quarter of its exports going to Turkey. Not surprisingly, they are also Turkey's best advocate in the EU. So a (Trump style) trade war with Turkey would probably significantly impact the poorest EU countries the most. They are already in conflict with the Western EU over climate policy etc. A trade war with Turkey would further aggravate the divisions in the EU. Yeah, the Western EU members could compensate (even more) the Eastern ones, but you can probably guess how smooth that's gonna go; see Poland. Basically, in this perspective the Western EU has the choice of paying Turkey directly, or starting a trade war with it and paying compensation the Eastern EU countries...
Besides, the EU's economy is not as booming as the US one. It still has major problems in its south and wants to avoid a recession. So the EU is probably looking for the least costly solution, rather than the "most muscular".
Also, even for Trump it was relatively hard for him to get the "small fish" Guatemala (one tenth of Turkey's economy) to agree to a Turkey-like deal (i.e. act as an off-shore asylum processing queue for the neighboring countries). Trump used mostly threats (of taxes on remittances and zero visas) and a few carrots (promises of more temp farmworkers visas). This level of bargaining won't really work with Turkey now, since they already accuse the EU of reneging their visa promises for Turkish citizens.
As a bit of an aside, the EU actually did impose some sanctions on Turkey fairly recently, in relation to the Cyprus gas fields dispute:
Today [27 February 2020] the Council has placed two persons under restrictive measures in relation to Turkey's unauthorised drilling activities in the Eastern Mediterranean. These persons are responsible for or involved in planning, directing and implementing offshore hydrocarbon exploration activities in the Eastern Mediterranean which have not been authorised by the Republic of Cyprus.
The restrictive measures consist of a travel ban to the EU and an asset freeze. Moreover, EU persons and entities are not allowed to make funds available to the two listed persons.
But as you can see these are more along the lines of the kinds that the EU has imposed against Russia (in relation to Crimea), i.e. they are not broad sanction. Their effect is probably going to be quite limited.
Also, as mentioned in the latter article, Council-imposed sanctions require unanimity...
Someone commenting below misread my answer as me saying that the EU must pay Turkey. I'm just saying that getting Turkey to comply via economic sanctions which is the topic of this question may not be as a easy as some would think. The EU presently seems to focus (instead) on a closed [Greek] border approach. The EU's response today is hardly similar to what it was in 2015:
Some humanitarian groups have expressed anger at how Greece and Turkey are dealing with the situation at the border, as Greek authorities said they had thwarted another 1,000 attempted border crossings overnight.
"We've seen quite a lot of unhappiness with some of the language that's being used," Euronews correspondent Jack Parrock reported from Brussels, citing von der Leyen's use of the term "shield". "They are concerned that this is language that is not appropriate from the European Union when a lot of the people coming are obviously fleeing wars in Syria and in other places as well".
"What we've seen today is a hard line and an inhumane response. EU leaders today have let Greece off the hook for closing its borders, and off the hook for shutting down the right to asylum for people in need," Eve Geddie, Amnesty International's Deputy Director of Advocacy told Euronews Now.
She said EU leaders had missed an opportunity to show solidarity with migrants. Disputing Turkey's classification as a "safe country", she said the EU should allow people to claim asylum.
Whether this border closure will prove tenable/sufficient in itself or not and then the EU might have to deal (again) with the problem further "upstream" in Turkey, either with carrots or (less likely) sticks... time will tell.
The EU has also been paying African countries to keep their migrants away... (sources on this vary between 1 and a 2bn euros) and more or less subtly conditioning aid on migration reduction.