No, communities on different instances will not have the same content. This is one of the features of the fediverse.

Each instance can have it’s own /c/memes and they won’t conflict. So, there could be https://lemmy.world/c/memes and https://lemmyverse.org/c/memes and they could each be different with different content and rules. If you want to see each instances ‘memes’ community, then you will need to subscribe to them individually from your home instance. Once you do that, the ‘memes’ community will be ‘cached’ on your home instance and it will show up in everyone’s feed on your home server.