So "saya" is only used once in such a sentence? What if I repeated it? And is my answer, even if less logical, just wrong?

You can use "saya" twice, for emphasis that you own both items, but it's not necessary.
"saya" functions as a possessive here, and refers to both items: "pakaian dan tas".
"saya" is the possessor, the owner of both items, the clothes and the bag.

So, yes you can use "saya" twice, both translations below are valid:
"Pakaian saya dan tas saya hitam."
"Pakaian dan tas saya hitam."

The same as in English, you can use "my" once or twice, both are valid:
My clothes and my bag are black.
My clothes and bag are black.

Ok, so I answered "The clothes and my bag are black" and it was considered wrong.

Your English sentence translates "saya" as the possessor/owner of only one item, the bag.
Only the bag is mine.
In Indonesian that would be something like this :
"Pakaian itu dan tas saya hitam."
"The/Those clothes and my bag are black."