You are right, but in German you also have to look at the gender of the noun:

So basically it's all depending on the article used: DER/DIE/DAS

DER and DAS use "Mein"

DIE uses "Meine"

An Example:

DAS Haus -> Mein Haus

BUT: DIE Häuser (plural) -> Meine Häuser

Another one: DIE Frau -> Meine Frau / DIE Frauen (plural) -> Meine Frauen

The dress translates "Das Kleid" in German , but it asks for the plural "Die Kleider" and therefore you need to use "Meine"

why is "Meine Kleidung" wrong?

I expected the plural of "kleid" to be "kleide" just like "hund/hunde". Is there a reason it's "kleider", or is it just irregular?

What is the difference between Mein and Meine? Don't they both mean my? Why is it that one is prefered over the other?

Depends on the gender of the noun, if it's feminine you have to use Meine, otherwise you use Mein... At least I think so :)