They can't "force" them to do anything.  Priests should go to jail rather than comply.

Now, this will make the perps more reluctant to confess, but I could see this wicked governor sending in shills with microphones to entrap priests ... just like they send under-drinking-age kids into various stores to see if they can catch them seeling alcohol to them.

So far there's no danger of a priest being able to comply, provided he uses a confessional and not face to face.

I don't know what's in a NO confessional these days, but we always had a black cloth covering the grate so the priest could never positively identify a penitent.

Of course they will try entrapment, anything to further their agenda of persecution.

Maybe it will go back to the old way where one could only confess to their parish priest.

In terms of the overall pressing needs of that state like drug addiction, homelessness, failing schools, cost of living, rampant crime, etc. it certainly speaks volumes that the marxist governor would focus on something so irrelevant to
the well being of the people. 
If you were to question the average person, the average blue haired feminist even, the seal of confession wouldn't even be in included in a list of their top ten thousand concerns. 
It's either some personal petty vendetta or the law seeks to set up some precedent for what's coming down the line
when noahide is established for the antichrist.