This sounds like a problem with iOS and not with Spotify.
It sounds like you have the continuity feature enabled. This allows you to also get your calls and texts from your phone, on your iPad.
The best thing to do is turn the iPad on silent mode (no sounds from alerts) and turn on do not disturb mode from the control center if you have this feature enabled so that it does not disturn the music playing.
I suggest looking at this support article from Apple that explains the continuity feature and how to turn it on or off: