"I am looking for my luggage."

Translation: Я шукаю мій багаж.

[VladimirYa15]

Please add another answer "Я шукаю свій багаж" related to that also.

[sagitta145]

Yes, to be honest, Я шукаю мій багаж sounds wrong to me.

Ти миєш твою машину. Він любить його кота.

Completely wrong. The last one even sounds like there are two different "він".

Is it actually a correct way to say it in general? I never did so in my life I think :)

[GeneM.]

I haven't learned about свій yet. Why is мій wrong in this sentence?

[sagitta145]

I have figured it all out now. It's not just ugly, it's grammatically wrong.

Same as in English double negative is wrong (I don't go nowhere), in Ukrainian "double pronoun" is wrong grammatically.

• If the noun that you modify with "my/his" is a subject, use мій, твій, його, ...:

Мій багаж тут. Його багаж там.

• If that noun is not a subject, use свій, if it's referring to the subject.

Я шукаю свій багаж. I'm looking for my luggage.

Я шукаю мій багаж. WRONG

Він шукає свій багаж. He's looking for his own luggage.

Він шукає його багаж. He's looking for some other man's luggage.

• If that noun is not a subject, and is not referring to the subject, same as the first category.

Я шукаю його багаж. I'm looking for his baggage.

Я даю йому його багаж. I'm giving him his luggage. (йому is not the subject)

[VladimirYa15]

In this case it sounds unnaturally and maybe even ugly.