According to the screenshot you provided, the first group membership is the unified group you created when you create the site which is not a SharePoint group:
As to you mentioned that "Site Settings>Site permissions>Site Owners group, I see that the site owners list (below) is a bit different and the Site members list is empty. The users in the Owners group are unable to add new users (no new button)", this is by design, we could not add SharePoint group member here:
Instead, you need to add members to group here:
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