I had to make sure to create an mlb the show account on the website other than the platform that I am using. You will have a MLB The Show account as well as your PS5, Xbox, or switch account. You have to get the link code for your account on PS5, Xbox, or Switch. You enter that in your mlb the show account. You have to link both accounts together.
Go to your account on the MLB the show website. At the bottom there will be a spot to put your link code in. Just be careful once you click link. It will have you choose between the new account and your current account for progression. Choose the one that you have been playing and it will connect.
Now retry logging into the app. It should work now. It worked for me.