So some news on this issue. I saw someone playing MLB the Show 22 last night online on a a stream via my PS5.
So I launched he game just now on my second "guest" account I have (when buddies come over to play COD & stuff) and it connected online immediately! I also tried the game on my switch using my daughter's profile and it seemed prompted me to set up an nintendo network id - so that may or may not worked there too.
It seems the issue is related to your profile/accounts and not that your system can connect. I tried unlinking and then relinking my switch and PS account from my MLB the Show account and that didn't help.
@Victor_SDS_MLBTS what is going on here? Please give us an update. I have also been messaging a Miguel in the technical support chat/email thingy and let him know what I found today.