Okay! So what finally solved it was a discussion with the WRC support. 

Within the Postman request, and the form-data; Content-type was specified as text/plain on each part of the form. That is why %request.Data("profile") wouldn't work at first. The content of each part of the form gets sent into the REST-service as a stream.
With this said, instead I had to use this code which finally worked:

set profileStream = %request.GetMimeData("profile")
set vProfile = profileStream.Read()

However, one thing worth noting is that when I did not specify content-type for each part of the form-data, then the original code I used worked: set vProfile = $Get(%request.Data("profile")) .

Thanks to WRC and everyone here who tried to help! :)

Okay! So what finally solved it was a discussion with the WRC support. 

Within the Postman request, and the form-data; Content-type was specified as text/plain on each part of the form. That is why %request.Data("profile") wouldn't work at first. The content of each part of the form gets sent into the REST-service as a stream.
With this said, instead I had to use this code which finally worked:

set profileStream = %request.GetMimeData("profile")
set vProfile = profileStream.Read()

However, one thing worth noting is that when I did not specify content-type for each part of the form-data, then the original code I used worked: set vProfile = $Get(%request.Data("profile")) .

Thanks to WRC and everyone here who tried to help! :)