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.