I noticed there was an option to release monsters, does this mean you can recapture them again?
Most monsters can be caught more than once using catch-and-release. Additionally, you can capture multiple Black, Blue, Green, Purple, White, and Regular Chocobos.
There are, however, a number of monsters you can only capture once ever,1 as Serge Bekenkamp's answer mentions:
- Golden Chocobo
- Twilight Odin
Retrievable using Mog's Manifestation/Moogle Throw
- Silver Chocobo
Note 1: Unless you purchase the Perpetual Battlefield DLC, which allows you to fight Valfodr. The drops for defeating Valfodr at his various levels include monster crystals for the normally unique tamable monsters.