Steam on browser problem

That's sort of famous bug. Try cleaning your cookies related to steam, disabling and then re-enabling any steam-related ad-ons and so on.

I even get logged out, just after a few hours
It‘s especially annoying as you need to enter the code from your phone every time

I cannot help but I can say I had this problem for a long time, but a few months ago it seems to have improved and is much less often now.

I have a worse problem where I cannot perform any community tasks in chrome like write a post, join a group or sell a inventory item.

It happens to me too before this. What I did to fix is to disable a rating script I used in SG (the old one). Now ESGST has that feature and it worked fine.

Well that's really interesting. Now that I think about it, it happened to me in the past, like all the time. Though, few months ago I reinstalled my windows 10 (clean install) and since then it didn't happened again. Haha, I hope I don't jinx it. Also to mention I am using Chrome, latest version and I am still using all the add-ons I was using before I reinstalled windows. Don't really know what fixed it, but I'm happy that it is.

Check softwares of clearing your cookies (Like Ccleaner).

just kill hola definitely, it helped me 100% with the same problem

Login, logout again, now login again but tick "Remember me" which will only appear if you are trying to login again.

Extend the expiry date of your session cookie maybe?

It's either cookies getting expired, or your IP is changing.
You using a VPN/Google Data Saver, this are the things that will keep changing your IP.