What social media are we talking? I access Lemmy through Tor every now and them. As for more intrusive social media (like Facebook and Instagram), they’ll try to associate your IP to your profile. So, they would be suspicious of continuously changing IPs, and ask for more verification (even more intrusive), or just ban you from their services. Also, they can easily know if the request is made from a tor node (try opening https://www.dan.me.uk/tornodes from a tor IP).
However, if I were to even ignore you accessing their service via a tor IP, it is still not private. Facebook, for instance, is very adamant on users providing their actual info. I tried opening a facebook account on multiple occasions, both with fake and partial information, and always got banned from their service. I have completely stopped trying now.
Let’s say you already have an account, and you are accessing the service through Tor (without them blocking or banning you somehow). It would still be a bad idea privacy-wise, because all your user activity is still being tracked, and linked to your profile (which might have your real info). So, you are still being extensively profiled, and your data would still be used for “improving their services and the user experience”.
So, unfortunately, the only thing to do is not provide them with more data points to track you, i.e. avoid intrusive social media altogether or move to social media which collect less data.
Then, to minimize the harm, be cautious of (third-party) cookies (reject them all) and may be use a different browser/session for Facebook.