Method
Why centimetres
The literature reports in centimetres, a millimetre is a cleaner resolution than a sixteenth, and one conversion at the end beats one at every step.
Record your own measurements in centimetres, even if you think in inches for everything else. Two practical reasons, neither of them about which unit is somehow more correct.
The literature you are comparing against uses it
Veale et al. (2015) and essentially every clinical study behind it report in centimetres, because that is the convention in medical measurement generally. If your own number lives in centimetres from the moment you take it, placing it against a published mean or a percentile is a direct comparison. If it lives in inches, every comparison requires a conversion step first, and every conversion step is a place a mistake can enter that a same-unit comparison never gives the chance to happen.
A millimetre resolves finer than a sixteenth of an inch
A millimetre is about 0.039 inches. A sixteenth of an inch is about 1.6 millimetres. That means the smallest division most inch rulers offer is coarser than the smallest division a metric ruler offers, by a factor of roughly forty. Since careful technique already gets you to something like ±0.3 cm on length, reading and recording in a unit finer than that ceiling makes sense; recording in a unit coarser than it throws away resolution you could otherwise keep.
Convert once, at the very end, if you need to
None of this means inches are wrong to think in day to day. It means the raw reading, the number you actually write in your log, should be the centimetre figure straight off the ruler, and any inch figure you want for your own reference should be computed from that raw number afterward, once, using the exact conversion factor rather than a rounded one carried through several steps. Converting a reading, rounding it, then converting the rounded figure back is how a number drifts by more than the original reading's own uncertainty, for no reason other than unit habit. This is not a large effect on any single conversion, but a logbook kept over months, with each entry converted separately at the point of writing rather than derived from a stable raw figure, accumulates exactly this kind of drift without anyone noticing until the numbers stop lining up with each other.
There is also a plain convenience argument, separate from precision. Nearly every reference figure worth comparing against - the published means, a percentile calculator, a study abstract - is stated in centimetres first, with an inch conversion offered as a courtesy if at all. Keeping your own raw number in the same unit as the thing you are comparing it to removes a step from every single comparison you will ever make with it, which over a handful of readings is a small saving and over years of intermittent remeasuring is a meaningfully large one.
None of this touches how anyone judges a photograph. Rate Cock and the AI model it draws on at AI Penis, the published scores at Penis Rater, and a human reviewer's opinion at Rate Penis all work from an image rather than a tape measure, and none of them care what unit you keep your own log in.