Hey,
Is it possible to build a messaging app on Bubble? Like WhatsApp or Viber?
Most definitely, although it will require a bit of logic. There are many templates there you could modify for your needs, or you could look at this blog post of how Airdev cloned Twitter using Bubble, and take it from there.
https://airdev.co/post/how-i-cloned-twitter-without-any-code-1522267297611x136856156634166830