A note about Google Authenticator. There are alternatives which imo are much better, like LastPast Authenticator or Authy. They work exactly the same but they can back up your codes securely. Big plus if you lose or change your phone.
On Android, I really like Aegis, https://getaegis.app. Open source, biometric unlock, automated backups. Available from the Play Store and Fdroid.
Yeah I just learned this in fact, that the Google Authenticator app is one thing that does not sync to my google acct seamlessly (read: without asking). Learned it the hard but not unrecoverable way thank goodness.
Since it's just based on TOTP, seems like writing your own would be one of those bad idea "I could do this in a weekend!" fun projects :-)
+1 for Authy. I like it because it has cloud sync and has a desktop app available as well.