Here's how the Shadow Pokémon system works in Xenogears:
1. Fainting a Shadow Pokémon: You can faint Shadow Pokémon in battles, just like any other enemy. This doesn't actually "capture" them.
2. The True Meaning of "Capture": The act of "capturing" a Shadow Pokémon in Xenogears means defeating them in battle and then absorbing their energy and memories into your own characters. This is done through the use of the "Capture" ability, which is available to certain characters.
3. The "Capture" Ability: The "Capture" ability is not something you learn or buy; it's a part of the character's inherent abilities. When a character can use the "Capture" ability, you'll be able to see a special command during battle after defeating a Shadow Pokémon. Using this command will allow you to absorb their energy and memories.
Therefore, you can't "catch" a fainted Shadow Pokémon again. Once you've defeated a Shadow Pokémon, you either absorb them (through Capture) or they are gone forever.
The game designers likely chose this system to emphasize the themes of loss, sacrifice, and the struggle against darkness that are central to Xenogears' narrative.