Just listen to your heartbeat and think again whether your cells don't track the passage of time.
Again. There is no counting inside a cell in principle. Only pattern matching of phisical molecular structures and "mechanical" triggers. No counters, no clocks, no notion of a number.
Do not confuse oscillations or rotation with time. Processes do exist. Time does not.
DNA
> What unit it uses?
That's defined, if rather complex. Essentially it's a basepair sequence.
> What is representation of the current value if the counter?
It's unary.
> How it deals with overflows?
It doesn't.
The notions of numbers, counting in general and the related concert of time required an observer to be established and formulated. These notions cannot emerge by accident in a purely mechanical perturbations the way life originated. There were no observer, leave alone designer.
This means that life could only use "concrete" phisical structures against which it does exact pattern matching, which is how DNA or RNA translation and protein sintesis works.
Message passing between the cells and triggering of gates and pumps are done by exact pattern matching for the very same reasons - there is no other way in principle. That is why there is no counters or clocks in the nature except human made ones.
Oh yes it has! it's arrow is called entropy and it ticks at the speed of atom decay.
There is no such thing as time, so Nature and evolution does not have any clock or counters. Life does not work the way we conditioned to think as observers. It cannot use abstract mental concerts which does not exist at molecular or cellular level. Cells do message passing and explicit pattern matching. No clocks or counters.