The tire on my bicycle rim slightly touches parts of my frame. Obviously, the rims are too large for my frame. However, is it possible to get tires with a lower "height" (see red arrow in the picture) to increase the clearance between tire and the frame (0.5 cm would be sufficient)? If yes, for what do I have to pay attention when buying a tire? Any other suggestions on how to solve this problem without buying new rims?
It's possible that your rims are not too large for the frame, you are just using tires that are too wide and tall than the frame was designed to accommodate.
Tire height is related to width, you can't specifically get a tire that is shorter. YOu just need to drop down to the next lower tire size. E.g., if your tires are nominally 32mm wide try 28mm tires.
There are only a few rim diameters in common use on new bicycles [559mm, 584mm, 622mm) it would be obvious if you had the wheels that were larger than the frame is designed for. Possibly though you have older 630mm rims a frame designed for 622mm.