I won a solo game with the Prelude extension 10 ++ times, but I didn’t get the badge (Solo 5).
Does anyone know what the condition is for it?

Edit: I need to pay more attention. :) "Terraformer" is one of the original achievements and should be triggerable. Sadly I'm afraid I don't know why it might not be triggering.

This and a number of other achievements showed up recently and I notice they are all marked "Legendary" with no completion stats. So they probably are not "achievable," so to speak, at the moment.

I have no insider knowledge, but I notice some of the new achievements mention Hellas and Elysium, expansion boards for the board game that are not in this adaptation at the moment. I have no insider knowledge but I'd guess there is an incoming patch with accompanying Hellas & Elysium DLC, and that the achievements found in the patch and/or DLC got added to the database ahead of launch.

Were you playing a single-player, offline "Local" game, or the "Solo Challenge"? The achievement is for the latter.

If it still doesn't work, maybe try it without Prelude enabled?

(And make sure your TM game is logged into your Asmodee account.)