Instagram appears to be doing a slow cloud rollout of multiple account support on Android, for those of you that desire personal and professional accounts and such. This comes ahead of iOS support.
Version 7.12.0 of the app has the functionality, though bizarrely, not everyone on this version will see it. If you're not on it yet, get it from APK Mirror or through the Play Store beta program.
To set up an additional Instagram account on Android, you'll see an option for it below "Clear Search History". You should be able to take it from there. Happily, an option to sign out from all accounts or an individual one is present.