I use "uber" when I really mean "lyft" - ie "Want to get an Uber?" is much more clear than "Want to get a Lyft"? I think any other more recognizable name would have been a plus for Lyft.
The only downside I can come up with is that "taxiing to" is more awkward to say than "ubering to". As Calvin says, though, "Verbing weirds language."
ps. I take Lyft 99 % of the time, for years already. Every industry needs an underdog to keep the big guys on their toes. In particular shady ones.