I have a problem, whenever I play a solo and pick some certain note in a string, some other string vibrates.

I've noticed they happen to be the same notes, for e.g. when I play the 10th fret of B string (which is an A), my A string starts vibrating.

Is there a way to stop this or is it an issue for every guitar?

It's really bugging me when I play, it creates distortion and I can't play the solos clean as they should be.

Please help, thank you in advance.
Uhm, learn to mute the strings you aren't playing. Guitars always resonate across the strings, it makes it sound fatter and, ya know, like a guitar. If you don't want this, learn to mute.
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thats string buzz when the string hits the fret and makes it sound bad.

You sure?
Quote by iPhantom
You sure?

Its not. That would be fret buzz. Completely different from another string ringing out.
Just practice muting like p o e suggested.