Yes, it has the same meaning, that your friend is vegan.

People tend to answer in the comments section because the design of the Italki questions section makes it easier to leave a comment than an answer

Thank you Phil and Yaser! I was wondering why people tend to answer here instead of writing your opinion in the correction section. I already chose the best answer, but when I ask a question people tend to answer here... I'm just curious!

As David says, they are *not* necessarily the same. “My vegan friend” may or may not be definite (although it is usually definite), but “a vegan friend of mine” is indefinite (implying you have more than one vegan friend). If we say “the vegan friend of mine”, it would be definite. It's usually a good idea not to choose the first answer as the "best answer".

Hi. Grammatically, both of them are correct. However when you say "a vegan friend of mine", you are actually emphasizing that friend among your relatives who is "vegan". On the other hand in the phrase " my vegan friend" there is no insistence on that characteristic. You just provide information about your friend.