Strictly speaking, GA is always 3rd party (it would only be 1st party if/when its used to measure google-analytics.com which is not a user-facing domain). For measuring any other website, its 3rd party.

I believe that this would limit GA's ability to store data in cookies on my system (user or session-based attributes). GA could still create unique identifiers based on my user or session-based attributes and store those IDs on their own servers, then call them back up when I revisit the site.

I'm still coming up to speed on this, maybe someone with more experience could correct or clarify my statements.