What metrics can be used to distinguish between capitalistic and socialistic economies? What macroeconomic indicators are meaningful to distinguish whether a country leans towards socialism or capitalism (and where to find such data, specifically for european countries)?
My opinion is that “capitalism” and “socialism” in this context is in the eye of te beholder. The European countries are mixed economies.

One standard metric is the size of government expenditures as a percentage of GDP. One issue is that different countries have different ways of measuring national accounts data.

The IMF World Economic World Economic Outlook (WEO) has a database that is standardised across countries. For any one country, the national accounts data would probably be a better source, but the key to the WEO is that they worked on comparability for you already.

Link to the data: link to IMF WEO data page

One series to look for is “general government expenditures as a percentage of GDP.” You want “general government” (and not something just like the central government) as countries like Canada have large sub-sovereign governments.

Expenditures gives us a rough idea of the footprint of the government in the economy. However, it will miss things like the effect of regulations. Instead of the government directly taking over some activity, it could force the private sector to follow a very specific set of rules. Whether that is “socialism” or a “properly regulated economy” turns into a political debate.

If you are only interested in European data, Eurostat would likely be your source. Eurostat has worked on getting European countries’ data to follow similar standards, so there is good cross-country comparability for now. I do not know what series to look for there, so you would have to dig. Link to Eurostate website.

The only other sort of data  to work with are various “ease of doing business” indices. The quality of those data series can be debated; there was some recent controversy with some World Bank data of this type in the news. The concern is that it is unclear what to include in these indices, and many of the inputs are qualitative. As may be obvious, I am not a fan of these series, so I do not have any recommendations (you can either search or someone else can give an answer).