Is there any possibility to auto login to sites via Facebook? When browsing Facebook via the iOS app one often browses websites through the app. Some websites need login and since the FB app doesn't support password handlers it's a difficult login process. 

So any suggestion about how to handle this the easiest way? 

I'd wish one could automatically open all sites in Safari at least if FB doesn't want to support password handlers.
This is a design decision by Facebook, and the best you can do is go into Settings → Safari → Passwords, where you can copy/paste your passwords into the Facebook browser to log into individual websites.

You can also specify a text replacement string for your email address under Settings → General → Keyboard → Text Replacements, so you can enter it more quickly into the Facebook app.