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?
Acceleration and velocity are vectors, which have magnitude and diretcion. Speed is the magnitude of velocity - it may remain constant while the direction of velocity is changing. What you are given here is the magnitude of the acceleration - if it is perpendicular to the velocity, the latter will change only its direction, but not magntitude.
Say the body has a speed of 30m/s, and initially moves in X direction, meaning its velocity along X is 30m/s, and along Y it's 0m/s.
An acceleration of 2m/s² means that the velocity changes by 2m/s every second. A centripetal acceleration is directed along a right angle to the velocity, so in our situation it's a Y acceleration, meaning that after a second we get a Y velocity component of 2m/s, while X velocity stays unchanged at 30m/s.
(Of course, this was a bit simplified, not giving exact results, as a true centripetal acceleration should change its direction a bit during this one second, as the velocity direction changes. But in our case, the resulting error is far below 1%.)
And that's what happens when a car moves along a 900m-diameter circle at 108km/h: its velocity changes by 2m/s towards the circle center every second.
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.