Actually once my friend was trying on some dresses and for one dress she used this sentence. Is this correct grammatically?
Unless you live in California, the "like" is extraneous. "It feels so cozy" is natural, where the "so" means "very".
In many parts of California (and possibly elsewhere), the local vernacular is to use "like" as a kind of punctuation. It doesn't usually add any meaning to the sentence, but it can add emphasis.
I told her that she was, like, being really stubborn.
That dress, like, looks so good on you!
Like, I wanted to go, but I didn't really want to go.
I wouldn't recommend using this, though. It's just information so you can recognize the vernacular if you hear it.
Only if like is an interjection, and it should be written this way to clarify:
it feels, like, so cozy
When X is like Y, that means means X shares qualities with Y. The implication in expressions using this interjection is the speaker cannot find the correct word, but is offering the first thing that comes to their mind or a simpler word to try to express what they want.
If like is not being used in this manner, it doesn't work. In like X, X must be a noun, and while you can say so like X you can't say like so X (though like so much X or like so many X is OK).