Misuse of Alike

The word alike is an adverb (even though it seems like an adjective), and shouldn’t be used to modify a noun; instead, use an adjective (perhaps same or similar).

My girlfriend and I have an alike hairstyle.

Because hairstyle is being modified, an adjective is required here:

My girlfriend and I have the same hairstyle.

My girlfriend and I have a similar hairstyle.

If you really want to use the word alike, change the wording of your sentence:

Our hairstyles are alike.

This sentence works because alike is modifying are, not hairstyles.