This is my imperfect understanding: While it is still possible to use Google Recaptcha, it causes a lot of privacy headaches, because Google processes personal data and places cookies. The latter means that you'll need at least some kind of cookie banner and make sure, cookies are only placed after this was explicitly allowed. But more importantly is the former: Google not only processes personal data, but it does this possibly in the US or somewhere else. This means - as I understand - that you'll need some kind of contract with them to protect your and your user's interests. That is usually a standard contract, however it is a legal binding contract.
And in some related cases of which I'm aware, courts ruled that you could theoretically be partially responsible if your contract partner disobeys privacy regulations.
While this is my best knowledge, there are of course no guarantees that my probably out-of-date-for-several-months knowledge isn't necessarily anymore correct.
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