So, why can't you cover (sing the songs made famous by) any singer you like? You can, if you don't force your voice to sound exactly like them. Most top singers do cover songs. But they do them their way. We already have one Elton, or Jerry Cantrell, or Maria Cary. We don't need another. We need a YOU! With that fact in mind, let's explore why you can damage your physical voice by trying to exactly cover other singers.
"If everybody has a set of vocal chords, they should be able to sound like any other singer. Right?" WRONG. The problem lies in the fact that everyone's' vocal chords and singing muscles uniquely different, much like fingerprints. They come in different sizes. Guys' vocal cords are approximately twice the size of girls'. This means that the sound of the female voice is a lot higher in pitch. (Think about the size of a flute, as compared to a tuba.) Then, to further complicate matters, vocal chords come in at least two sizes for both girls and guys large (low sounding) and small (high sounding). The quality and range of voices are governed by their size and the size and shapes of the neck, sinuses and other parts of the body. It's usually very difficult, if not dangerous, for guys to sing girl's song - or visa-versa in their original keys without pushing and straining. These songs usually need to be put in another key that is best for your voice. The most important thing to remember about changing keys is that you need to sound the best you can. Singing a song in the wrong key for you will not do that.
So it boils down to this: you'll never be able to sound exactly like another singer, so stop trying. You might consider emulating some of their unique style (the way they pronounce words or drop the ends of phrases, or growl, etc.) but that's as far as you should go unless your natural sound is very close to theirs. We have a "them" We need a "YOU".