Additionally, like many social sites, the idea is to collect user-created value for free, centred solely within their control. That aspect echoes of the great problem with scientific publishing: Scientists produce information, and give it away to some corporation for free, which then charges the public to access it. The user comments, even if informative, are stuck on the site; you have to return to the site to access those comments.
Even if "All team members have an academic background and share the desire to make science more open and widely distributed" [0], there is no guarantee that the user-populated database of annotations will remain free or accessible in the future, since again, it's stuck on that site.
As for the online part you may be conflating your point about offline annotation. Fermat's Library is about annotation sharing: implementing this functionality without any sort of identity management appears to me that it would basically be asking for trouble (imagine you could send e-mails to anyone without logging into anything).
It's easy to casually throw around phrases like "log in to a proprietary service" to criticise something. Perhaps that sentiment is understandable, but it doesn't really move the conversation forward or lead to better solutions. In this case I think it's just simply missing the point: at least at this point in time Fermat's Library's mission doesn't appear to be about grabbing your identity for profit. So unless there is something particularly fishy about what they are doing, that sort of criticism seems unfounded.
What would be the "first-party software" in that case?
That said I wish Fermat every success. Any way that helps you drink from the firehose of publications is a much needed step forward. Even if there is so much more you can still do in this respect...