Square got the rights to use Mario enemies in the game, correct, but some of the Mario enemies in the game use completely different names because they probably couldn't get the rights to use their names in the game, or didn't want to pay for their names, so they made their own names for some Mario enemies.
The Beezos (The Fly Guys from SMB2) in the game are called "Shy Aways" because Square possibly couldn't get the rights to their original names, hence why they use that name.
I guess Square didn't get the rights to use Koopa Troopas, so they made their own Koopa Troopa species and designs for them, and their names go under "Terrapin" for the green-shelled Koopa and "Terra Cotta" for the red-shelled Koopa.
The Shy Guys on pogo sticks in the game are called "Shysters", but since they had the rights to use "Shy Guy", I doubt this one was an issue.
These are Bill Blasters, but they're called "Blaster" and "Big Bertha" in SMRPG.
Lava Bubbles are called "Sparky", probably because Square didn't get the rights to their OG name.
Cheep Cheeps go under the name "Goby", and you basically get the rest..
And as for the alternative names of the Mario enemies, that means those names are owned by Square, which is why we haven't seen those names on a Mario enemy ever since Super Mario RPG.