Black-Scholes with its assumptions results in a unique price for the call. If we introduce stochastic interest rates, would this still remain the case?
No, if there are no claims traded on the stochastic interest rate because then the market is incomplete.
Yes, if claims on stoch ir are traded because then the market is complete (all risks can be hedged) and there is a unique risk-neutral measure. Recall that in fact a zero coupon is a claim.
The latter ("yes"), in theory, is by far the more common situation. But in practice there is always some form of market incompleteness (we just make it complete in by reducing the number of variables in our models and/or ignoring other risk drivers in our models).