Does the size of a car engine affect the acceleration?

Kinda-sorta.

An engine's power is determined by two factors, the turning force the engine can create and how fast it can spin for a given amount of turning force.

To get good acceleration you need a bit of both. Amazing turning power but only one turn/minute wouldn't work particularly well. Nor would an engine that'd spin like crazy but very weakly.

Generally, bigger engines are better at turning force and poorer on spin, while smaller engines are better at spin but poorer at turning force.

Now, obviously there are things you can do with gear boxes, to turn slow and strong into fast and a bit weaker, or fast and weak into slower and stronger.

But usually, for two engines with a similar power(horsepower or kW) rating, a bigger engine is better for acceleration.