For example. Would it be wrong to say:
My hobbies are singing, shopping, soccer, and stationery.
Why or why not?
Pretty sure you mean stationery; otherwise you're also including an adjective. As far as your actual question, singing and shopping are also functioning as nouns. They're gerunds, formed from verbs, but they act as nouns. (There's always some questioning as to whether you can say that a gerund is a noun. But certainly we use them like nouns, as you've done here.)
As a matter of style, I would try to have all the elements in a list be parallel. Especially if you start it with two gerunds, you're setting up an expectation, so it would work best to continue with gerunds (playing soccer, and whatever you do with stationery—making, collecting?). Otherwise, I would at least mix up the order. Maybe:
My hobbies are soccer, singing, stationery, and shopping.
But those are just style suggestions. The way you have it is not ungrammatical, and frankly, the style wouldn't even really stand out to me that much. I'd probably note it in passing, but it's not a significant issue.
Of course, my hobbies could be tennis, soccer, football, puzzles, music, piano.
You could say it should be playing tennis, solving puzzles, listening to music, but the above is just fine.