Do the units for acceleration break down when we're talking about bodies undergoing centripetal acceleration at a constant speed?

I've always taken values like $$2 m/s^2$$ to indicate that the speed of the body increases by $$2m/s$$ every second, but I fail to see how this can apply for situations where we're talking about bodies undergoing centripetal acceleration (at a constant speed). Yes, I agree that the velocity is changing, but its magnitude isn't and it feels, at least to me, as if the units break down here: if $$m/s^2$$ suggests that a particular $$m/s$$ value is being added to the initial velocity each second, how does that apply for bodies undergoing centripetal acceleration at a constant speed, where the speed isn't changing?

Even if you have $$a=2m/s^2$$ for time t=0 it does not say that the velocity changes by 2m/s in 1s- this supposes that a is constant, if a=2m/s^2+2m/s^3*t the velocity change is only in the very first moment $$2m/s^2*\Delta t$$ Same with centripetal acceleration, it changes not magnitude but direction all the time, so it alters the direction of v continuously.