16 isn't particularly old for a cat, it isn't all that uncommon for cats to live to 20 years old or so and still be fairly healthy. Pretty rare for them to make it over 20 though before getting significant problems which require medication (usually kidney problems and/or diabetes) or they get cancer. Not that different than what takes out most people as they get up to 90+.

In any case, it is hardly very surprising that all the 30+ year old cats disappeared now that we use microchips to track them. The current world's oldest documented cat is only 26 which shows how ridiculous the claim of a 38 year old cat is.

If the guy wasn't a liar he may just have been senile and just got confused. Vets don't see cats often enough (except near end of life) to be able to tell two similar looking cats apart without a microchip.