Yes, you need an MLB license to make an MLB game. Obviously that goes without saying. When I say anyone can make an MLB game, I mean there is no exclusive license deal that exists for other sports. Anyone can make an MLB game.
Let me break it down for everyone as someone who spent time with that studio, especially the licensing and authenticity sides of things.
MLB had different structures than the NFL at the time. EA bought the sole sim license. Nobody else can make an NFL "simulation" game.
MLB was structured like this:
Since the beginning from 989 sports to SDS, Sony had "first party licensing" with the MLB and another with the MLBPA.
Those contracts were on a 7 or 9 year renewal cycle. I can't remember the exact dates of the two.
Other publishers were able to in that timeframe, purchase 3rd party licenses and continue to make games. 2K tried (locking out EA) for a while, but eventually bowed out with failure after failure. MLB The Show was the "go to" game.
Microsoft
(who owns the High Heat IP which they purchased and vaulted to entice EA to their platform and on Live back on the original Xbox first gen) could have also purchased a 1st party license to utilize that IP for Xbox gamers... they never did.
MLB then decided to make a third party game themselves with revitalizing the RBI Baseball IP (after 2K did not renew), funding their own dev studio, that only lasted for about 2 or 3 years, and they too bowed out. MLB The Show was the "go to" game.
Sony's license was up for renewal. The sole 1st and 3rd party license was up for grabs to give Sony full exclusive use of the license (like EA has structured with the NFL for Madden) and thus you see what we have today with a mutual agreement and to build more revenue for both parties with their version of MTU/FTU with Diamond Dynasty.