Hello
When you were on an exchange account, you were using Teams for work/school edition, which is basically the business version of Teams. When you use a personal Microsoft account to sign into Teams, you are on the personal version of Teams. It may appear to be the same Teams app, but different types of accounts determine the version of Teams it gives you.
According to the error you received, it sounds like you are joining meeting hosted by organization user, is that right?
In this case, it looks like they do NOT allow personal users to join their meeting, which can be changed via their meeting policies.
However, you can also use this alternative to join such meeting. Copy and paste the meeting link to the browser inprivate mode, when prompted to use the app, do not use. When prompted to sign in, also do not. You should be able to get into the meeting that way.