I used my SE login to log into my account on my mobile browser, and I have not had any trouble using the drop-down menu while on it to access my different communities. The drop-down menu "knows" that I am logged in.
However, I recently noticed, while on Programming and Puzzling SE, where I do not have an account, that the drop-down menu treats me as logged out. I checked one or two other sites where I do not have an account and ran into the same behaviour. At the same time though, I am still able to access my inbox and reputation tracker on these sites where I do not have an account.
Note: I don't have this problem on the desktop site though: I can see a logout option and my list of communities on the drop-down menu even on sites where I don't have a profile.