Is it okay? And besides the different colors I still expected better photos in general, i don't understand if my camera is faulty.
If OK is the only question then it's OK. Color profile has no objective answer, from flat to real to saturated they're all subjective. At least on 13T you can switch between the authentic and vibrant profile, as well as the other available filters. At the worst, the Pro mode is there for manual tuning, especially try changing the white balance, it's the most influential.
13T series is NOT a flagship series, though, it's an upper midrange instead and you're capturing indoors, which also means low light. Therefore, a combination of lower quality sensor, lens, ISP as well as algorithm it won't match the true flagship the 13 series without T. It's just how the market segmentation works, i.e. I would be very angry if my 13 Pro result has a quality matching your 13T Pro, why should I even bother buy the flagship?
So no, your camera ain't faulty, it's just its default color profile doesn't match your taste.
