Is it a professional obligation to have to state a political position which pronouns are in our society?
I don't really understand what you are asking, but there are times when there will be a professional obligation which applies to those in a professional position which might not accord with personal views.
So a GP may have a professional obligation to provide a pregnant woman with impartial advice regarding termination of a pregnancy even if they oppose abortion.
A solicitor may have a professional obligation to act in the best interests of a client in a divorce case even if they personally oppose divorce.
A teacher in a pastoral position may have a professional obligation to act in a manner that is respectful and understanding where a pupil was come out as gay even if they personally think homosexuality is sinful.
The point being that the professional obligation are set by that profession with an expectation that everyone, during the time they are practicing that profession, should adhere to those obligations.
What they do, say or think in their free time is up to them, albeit with the proviso that publicly voicing certain private views may undermine your ability to act in a professional capacity or even bring your profession into disrepute.