As a long since retired metallurgist I am absolutely sure that a plastic bowl in a stainless steel sink would not, of itself, cause the rust spots. However, (there's always a however, isn't there?) it is possible that something, for example, tea, that has been in the bowl and then tipped down the sink could be the cause of the rust spots. If the rust is confined to the waste outlet area it is possible that this part of the sink isn't actually s/s but is a chromium plated insert and this will certainly, over time show some pitting if tea / tea bags are allowed to remain in contact with it for any length of time.
In any case, any rust or discolouration can easily be removed by light rubbing with a green scouring pad with a dab of Cif. Even a small drop of Domestos, well rinsed off, will work wonders.
But a plastic bowl causing rusting? No, Never!