Trademarks are granted within domains. Uber-the-dickhead applied for a mark in the domain of "transportation services" (not sure that's the exact name, but they are all quite vague), whereas Uber-the-Floridian applied and got one in the domain of IT.
This happens a lot: For instance, there are two entirely independent companies using the trademark "Kenwood": a UK one, making kitchen equipment, and a Japanese one, making electronics and radio communications equipment. An American butter company also trades freely as "Finlandia", two aisles away from the better-known vodka brand.
Canon EOS cameras are in a different domain to Volkswagen's poor-selling EOS Golf-with-a-metal-roof, so both names were granted. "Dove" is granted to Mars and Unilever for a chocolate bar, and a range of soap products, respectively - to further avoid confusion, Mars uses the "Galaxy" name for the chocolate in countries where the soap is popular, but "Galaxy" is also a trademark of Ford, although not for chocolate... meanwhile "Puma", one of Ford's other names, is also a globally-known brand of athletics gear, and so on. Basically, without domain limitations, there'd be very few good trademarks left by now.
The problems happen when a company that starts in one domain, then grows into another, or when a company gains such a bad rep that it defames the name of everyone else using that name.
Uber is that second type, but a famous example of growing into infringement is Apple: the trademark "Apple" was given to the computer company, despite already being held by a record label (The Beatles' "Apple Corps") because in 1976 (and now), these were different domains of business. In the 1980s, when Apple added audio recording and playback to their products, the record company secured a legal agreement preventing the computer company operating in the music business, which didn't seem like a problem for Apple until they started selling music on iTunes.. In the end, lots of money changed hands, and everyone is sort-of friends now, as evidenced by the availability of all the Beatles' albums on iTunes, but Apple-of-Jobs did need to get permission from Apple-of-Beatles for the right to use the name "Apple" in connection with selling music.
[Trivia: the Mac alert sound, "Sosumi" is a reaction to the original, 1980s-era spat between these two. The name of the somewhat Beatles-like organ chord was originally called "Let it Beep", and then changed to "Sosumi" in reaction to complaints from Apple's Legal department. The developer passed it off as being a Japanese word, but it's actually three English ones ]