I raised the same question on GOG forums: https://www.gog.com/forum/geneforge_series/does_geneforge_1_mutagen_support_achievements_like_the_store_page_says
I asked GOG support about this a couple of days ago but no answer so far. I just contacted Spiderweb support as well. In summary, the store page mentions achievement support and your GOG profile shows completion percentage for the game (always 0%) but no achievements in the list. I played the game a bit on GOG and unlocked the first in-game medal (completed the tutorial) but no GOG achievement. I tested on both Windows and Mac.
Looks like either the store page is inaccurate for now, or the achievement list is still missing/hidden on GOG. Also, the Steam version had an issue unlocking achievements (already fixed) so it could be that as well for the GOG version.