For a lot of people if not everyone , their belief lies in reputed and "trusted" companies.
This shouldn't be the case and especially so with an authenticator. Any particular reason why? I have been using authy and I say it is any day better than Google authenticator, for it doesn't BACK up your fucking data. Almost had a nightmare when I un-installed it by mistake, had to contact so many exchanges to remove the 2FA 
I personally have never had any issues with authy.
I personally have never had any issues with authy.
The main reason why I wouldn't use Google authenticator or Authy is because it's not open source. There's open source alternatives which have been mentioned above which are simply better. Before you say Google authenticator is open source, the app which you download on the store is not.
Also, for their multi device feature to work, then your keys need to be stored on a server for it to work. So they certainly keep records/data. These are encrypted by the backup password you set. This is encrypted using AES and a randomly generated salt using PBKDF2. So the data is stored otherwise, it wouldn't be possible to restore access using another device.