"What kind of shirt are you looking for?"
Translation:Millaista paitaa etsit?
Millainen and minkälainen are synonyms. Both should be accepted as there is no difference between them. Note the spelling of minkälainen which reveals (hint: vowel harmony) that it is actually a compound word of minkä (of what, gen.) and lainen (sort, type). Millainen is also a compound, but I do not know where the specifier ,the first part, comes from. It may well be a very old form.
All interrogative pronouns ,the question words, are nomials in Finnish, i.e. the declinate the same way as nouns, adjectives and all other pronouns. The question word is the object of looking for, so it must be in a grammatical case that shows that it is the object, here in the partitive.