Why OMNY card did not show on transit card in wallet on iPhone 13?

sberman's answer is correct.

Your card doesn't show up on iPhone so most likely that transit card is not supported by Wallet.

If you'd like to confirm you need to contact the issuer of the card which in this case for the OMNY card you would need to contact the Cubic Transportation System who is the company that issued that card and ask them if it's supported on Wallet and if not if there are any plans to do so. If they don't have an answer then try with the MTA and see if they know.

From this article:

Where you can ride transit using Apple Pay - Apple Support

”If you don't see the transit card you're looking for, contact the transit agency to confirm that the card is eligible to be added to Wallet.”

Simple answer: No!

But OMNY readers support contactless cards.

OMNY: About Contactless Payments

Next time, perhaps if you show a more cooperative attitude, folks will be more receptive to assisting you.

Really? Have you confirmed the card is eligible to be added to Wallet?