Like others have said, if you log into a website that already knows your identity, you are not anonymous. They already know who you are.
But if you are talking about a different scenario in where you anonymously access another website with the same VPN IP, then read below.
If you use the same VPN IP with Website X that knows you (for example, your university) along with another Website Y that doesn’t know you (for example, this forum), it’s “possible” that you could be “de-anonymized” on Website Y, or that your anonymous browsing habits on Website Y could be known to Website X.
Maybe Website X’s fingerprinting of you was somehow leaked or shared or something, which allowed Website Y to crossreference it with its own fingerprinting of you. Or maybe Website Y’s fingerprinting of you was leaked, which allowed Website X to crossreference with their own. It’s hard to say because this is not my field of expertise, so take this example with a grain of salt and wait for the fingerprinting/tracking overview to be completed.
But either way, VPN IPs are usually shared with many, many people. It’s probably unlikely that crossreferencing only VPN IPs would work in “de-anonymizing” you (home IP addresses would work, though, since they are unique to your home network, I think). Not only this, but IP addresses are only one aspect of fingerprinting. There’s a lot more to fingerprinting than IP addresses. Fingerprints could be successfully crossreferenced even if you use different VPN IP addresses on each website because you could share many other fingerprinted aspects, like GPU or window size or operating system or whatever.
If you do not want to be deanonymized between websites, use Tor or Mullvad. But again, as others have said, using Tor or Mullvad on a website that already knows who you are (like your university) will not make you anonymous because they already know who you are.
Edit: I have not checked these out specifically, but they may be of significance to you: