People have asked me before if I'm about "to uber" or "take an uber" someplace and they say it in an obvious way that implies "any ridesharing company" (or lyft in my case since most people know I only lyft nowadays).
Uber just as a word for ride-sharing has become ingrained and won't be easy to get rid of, IMO.
That's more due to the ubiquity and dominance of Google itself.
It's rare to hear someone say "I Googled it on Bing" or even "Let me Google my email" when they're using Outlook. Maybe not unheard-of, but definitely nowhere near the threshold needed for genericization.
True AFAIK but if you ever give computer support you'll find people "just google it" and use the greeting page on their browser [aka "the internet"] which is just as often Bing or Yahoo as it is Google. Google, the verb, is definitely generic but the RTM holders of Google have several hundred million of $currency to spend on lawyers to say it isn't.