I have a wide range of hobbies, I love doing sports, cooking, learning languages, shopping and so on. Widespread has an entirely different meaning. It would be wrong here.

You could say "my hobbies are widespread" ;)

No, you can't say 'my hobby is widespread' if you mean that you have a lot of hobbies. That is not what 'widespread' means. 'Widespread' means that you find a particular thing in a lot of places or that it is common to a lot of people - for example, 'This problem is widespread among young people' or 'This disease is widespread in many poorer countries'. If you were to say 'My hobby is widespread', I would assume that you meant that you had one hobby, and that many people did this hobby in many different places.

I have many interests. I love sports, cooking, languages, shopping, and so on. "Hobby" is becoming less frequently used in English. It seems to refer back to a time before the internet when people made things with their hands. Of course, it's still fine but I'm not sure that it includes shopping.