Hi :)
Yes, the phrase "waking up to X" means that X is the first thing you experience upon waking. So, "waking up to you" means "you" is the first thing the singer is aware of after waking up.
The "up" in waking up basically suggests a result to the action. It's not exactly the same as a perfective verb in Russian, but it's close. It's a very subtle difference, even for a native speaker.
Also, you may have heard the phrase "wake up to yourself!", which we use to tell a person to become aware of their faults/situation.