How do I train my singing voice to sound exactly like a specific artist? Like if I play a song of his then turn it off and sing that same part my voice sounds exactly or almost exactly like his.
Imitate. Sing the notes he does, with the inflexions he uses. Sing with a favourite recording then obtain a backing track, sing with that. YOU'LL never be satisfied. No-one ever is satisfied with the sound of their own voice. But maybe, after a bit, your friends will start to say "You sound just like him!" And that's as close as you're going to get, be satisfied.
Aim for the WAY he sings, not so much for the sound he makes.
Of course, if you're a teenage girl and your model is Barry White, you've got problems :-)
I am curious for the reason for imitation, but will answer regardless. If you want to sound similar the most efficient way is to record your voice and compare it to the original sound.
The higher quality audio playback the better. Listen closely for tone, articulation and timbre of both voices. Adjust your voice to match what you are hearing and eventually you will notice progress.
Shape your airways in the shape of his in order to match his formants. Larger mouth and dexterous voluminous tongue provide you with additional flexibility in modeling other mouth shapes.
Your voice sounds different internally and externally, so you need to work with headphones for matching your own and the target voice. A delay of a few seconds will help alternating vocalizing and listening without having to fiddle with your equipment on a constant basis. Time-spectral analysis may sound nice to have, but figuring out something that will work interactively is pretty hard.
Try getting a close look: using the same kind of microphone is typically a good idea since its qualities will usually be a relevant part of the aggregate sound.
And you always need to ask yourself the question "why?". Mimicking somebody else does not make use of the full potential of your own voice. And the perfect mimicry is playback.
If your general reinterpretation of a piece is unpropitious compared to the original, your imitation of the original is scarcely likely to be better.