It’s less that Twilio specifically owns it than problems resulting from corporate ownership. Briefly:

1. You can’t get your data out of Authy. Actually you can, but it’s a long annoying process involving installing an out of date chrome extension and using developer tools.
2. Privacy issues. Authy links a lot of data including location to your identity.
3. Authy supports SMS account recovery (which is inherently insecure) and doesn’t allow users to disable it.