.eu's not huge in europe. But it's not .biz either.
Some customers might not notice, for some it'll suggest you're heavily EU focused - be aware .fr, .de, etc are definately a thing, and some might mistake you for a government agency offshoot.
But I don't see it heavily associated with .lol, .info, .biz, etc.
The other side of the coin is that non-.eu customers may associate your products or services with only the EU, or perhaps feel excluded.
Go with EU or a ccTLD(s) in Europe. If you are under a ccTLD where the registrar is rather strict on who can be assigned domains you might even score higher in search engines (or at least not get penalized).
COM is fully under US jurisdiction (though really most of the DNS system is).
Better to have the exact "servicename" name, so would go for the .eu but buy both. That said, surely there are more options than only .eu and .com?
Bare in mind that in order to regiser a .eu domain you must be an EU citizen, resident in the EU or your organisation must be established in the EU. (or the EEA - Iceland, Liechtenstein and Norway)
And yes, they will take away domains if you don't continue to meet the requirements - it happened to me (thanks, Brexit).