Am I allowed to pick and choose which marriages I recognize also?

You may want to repost this from a different source. The site this article is posted on is explicitly anti-LGBTQ+.

Why did you link to a right wing website? This appears to be a radical conservative Catholic site. Gay people's marriages are "marriages." Transgender people are "transgender." Seriously. Look at this fascist horseshit:

The legislation also states that private people and groups aren’t bound by the 14th Amendment to the U.S. Constitution, which guarantees due process and outlaws discrimination at the state level.

Empty right wing virtue signaling. There’s absolutely no way a measly state law that says “this special group of US citizens is immune to the Constitution” can be taken seriously by any reasonable court.

Fuck everyone who voted for this.

Wasn't that ... always allowed?

would allow private citizens and organizations to refuse to recognize same-sex marriages without facing punishment.

What punishment did they face previously for not 'recognizing' a marriage?

...reaffirming that marriage, as a concept, is entirely useless.

If you need to make what you already have ceremonious in some way, just do something else that's not centric of government and religion.