Uber is available in Cologne. Those answers provided by google should always be taken with a hefty pinch of salt - if you click through you'll see that the source for the quote in your first screenshot is a post on a trip advisor forum from "5 years ago" (exact date not displayed) and the quote in the second screenshot is from an article written in October 2018.
As for where Uber is available in Germany more generally, their website has a list of cities where they operate. Currently in Germany that is:
Note that Uber's model where essentially anybody with a suitable vehicle can sign up as a driver is not legal in Germany. In Germany it's essentially just a platform where you can reserve normal taxis operated by licensed taxi companies. They still have Uber's demand-based variable pricing model rather than the usual rates (set by local authorities) which apply if you catch a cab yourself.
Whether Uber can be recommended is probably a bit opinion-heavy for this site, but for me: I don't see any particular benefit. They're basically normal taxis except the pricing is less predictable. But if you don't speak German, I can imagine the app might make things easier.