On mobile devices in particular, different apps (such as Facebook and Twitter) use different browsers. While it is necessary to log in for each app/browser, if you leave the “Stay logged in” box checked, your credentials should be saved in that particular app/browser.
If you are on a shared computer, you should avoid storing your user account/password in the browser. However, if it is your personal computer or a secure individual profile, you can allow your browser to save your login information. Similarly a password manager, such as 1Password, Dashlane or Last Pass are also secure alternatives.
See also: [ What is the domain I should save in my browser? ]