Why we can analyse only 5 or 6 games per day now?
I am also wondering same. Frustrating.
A huge number of games are being played on lichess - and a lot of those are beginners who care to use the computer analyisis features instead of doing their own analysis of the games - (not to fault them, you have to have a certain level of ability to guide your own analysis of a game). As such, the hardware is probably just under a heavy load and so it has to be divided up between people somehow.
Just be glad it's not a paid feature or anything like that.
Is it possible to somehow replicate this by running stockfish on my local PC?
To some extent, though you the time it takes to complete an analysis will be quite long - Though all of the "convenience" features (like designating moves as mistakes and blunders, or the "learn from your mistakes" thing) will need to be done manually.
You can't post in the forums yet. Play some games!