Yes. Your game is tied to your Discord account, not to a server. Your Pokémon, ₽, bag and Pokédex come with you to every server that has Catch 'Em Cord, so you can catch in one server and battle in another.
What belongs to a server
Wild Pokémon spawn from each server's own chat, so what you can catch depends on where you are. Legendary chases and the weekly Server Challenge are per server too.
Trading and the market
/trade is with someone in the same server. /market is shared across every server, so a Pokémon listed in one can be bought from another.
Want it in your own server?
Here's how to add it. It takes a few minutes and your Pokémon will already be there.