Are you using this site and Lemmy in general from a desktop browser?

I just did the test:

• configure the home instance to lemmy.world on lemmyverse.net
• click on a community on lemmyverse.net, in this case [email protected]
• this opens the community in a new tab with https://lemmy.world/c/[email protected], so that I am actually logged on lemmy.world, instead of opening https://lemmy.ml/c/memes

Is this not what you were describing above as desired behaviour?