The concept of F=Ma is an ancient one but often seems a bit arbitrary. Why should this quantity of acceleration define the conserved energy and momentum? It can be thought of classicaly in the sense that velocity does not change without impulse. Thus, the most simple unit of change would be an acceleration. The factor of mass refers to considering the net change, because each particle is one piece. The fact that the sum of accelerations over time or space, as E or P, is conserved, comes from velocity being conserved, or transmitted as 1:1 between equal sized particles. Quantum is even simpler to imagine, each force is just the impulse of a photon divided by the objects mass. Of course photons are conserved.

It helps to think of these discretely. I hope this helps.