Any model, I downloaded, have no textures. I guess CAD models don't support textures at all, right?
Depends what files you will download...
IGES are ok
choose only files saved in the software you work with
and last option: activate realview in solidworks and a new world will be reviled to you :)
Will not it have a texture, if I open it in a different software that supports a same format?
If you export a model in some universal format, such as STEP, STL, etc., it will most likely not have texture, for compatibility purpose.
If you open files on a specific software that allow textures, there might be textures available, if the author have set it up properly. An example is this CAD model of Raspberry Pi 3 Model B Reference Design Solidworks CAD Rpi Raspberrypi Raspberry-Pi, which have textures if opened in Solidworls Format. Opened on STEP, STL or Parasolid, there is no texture.