To really copy a song takes a lot of patience and work, perhaps more than composing something original. Unless you were blatantly ripping something off, it's unlikely that you are going to "copy" it, so don't worry about it too much. Your favourite composer was influenced by some other music that they emulate aspects of without copying. Genres exist and songs within them sound similar, without copying each other.
To answer your question, my solution is to listen actively and take notes. It's one thing to hear something and like it, but another to be able to describe why you like it. Listen, close your eyes, open them to write an observation, repeat. Now you have a handful of ideas that you can re-interpret in your own way.
Also put the inspiration song in your DAW and use it as a reference as you compose. It will help you notice things like arrangement, instrumentation, and give you a target to work towards.
I'm Canadian. I went to a concert in Michigan (Third Eye Blind) and half the crowd had some article of clothing with the American flag on it. Not really that weird, but it's more than I'm used to seeing Canadians sporting Canadian flag clothing.