When composing music, how to be influenced by a soundtrack (of films, video games...) without copying?

Deliberately copy snippets of a work you're interested in as a study -- e.g. transcribe it -- and experiment with elements you find interesting (rhythm, chords, synths, effects, whatever) in small test pieces to make sure you understand what's going on. Let the ideas stew for a while and then much later try to use the techniques you learned in a real piece.

That's what I do anyway.

Breakdown your favorite songs by chord on a keyboard until the relationship between chords and moods clicks, then you can recreate feelings rather than duplicate the mere sounds

I think that's a very difficult question to answer. But if you want to be as close to foolproof as you can, consider researching prior cases, such as that of Vanilla Ice and Ed Sheeran.

" An artist never works under ideal conditions. If they existed, his work wouldn't exist, for the artist doesn't live in a vacuum. Some sort of pressure must exist. The artist exists because the world is not perfect. Art would be useless if the world were perfect, as man wouldn't look for harmony but would simply live in it. "

Andrei Tarkovsky

Listen to it a shit ton and then when you go to make/mix dont listen to it, and let the memory of the thing in your brain speak to you. Thats how I do it anyway. Since human memory is full of holes I usually get something different but similar.

I read Jeff Tweedy’s book on songwriting and he talked about borrowing themes from anything and every thing. Like that chord progression? Use it. Interesting cadence or rhythm? Borrow it. It’s not really copying, but more so taking inspiration from other works. It’s always good to keep your ears open to try to find things in the world that resonate with you. I often sit in my back yard and listen to the birds sing and imagine what a bird song would sound like applied to something on guitar or lyrical cadence.

When I'm writing, if I just listen to the songs I want to take inspiration from the day before a few times (properly listen, with intent) the main ideas/vibe of the songs are far more present in my writing then usual

The secret is getting bored with it while copying so it turns into something different.

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.

Basically by letting it linger in your mind and then when you make a song you remember certain things about it and some of those things you try out.

Now legally it gets complicated, but it always comes down to how close you are to the original.

Listen to the soundtrack, then compose something new

You just copy, but don't from a lot of places. If you only copy from a single thing your work is a simple derivative of that one thing. If you steal from 20 sources, that work is a masterful blend of multiple influences.