That is a new one to me. I've never seen a DS3231 clone, but the DS1302 is known to be cloned, that's been going on for years. Obviously, it's harder to make a DS3231 because the crystal has to be embedded in the package. Did they maybe combine a DS1302 clone with a crystal somehow and falsely label it? Hmmm.
A real DS3231 shouldn't lose accuracy with high temperatures, as it is temperature compensated.
1 minute per month is less than 1.0 PPM and is a reasonable expectation from a single chip TCXO. It would be hard to calibrate it for any higher performance than that, unless there are absolutely no environmental changes. Eventually it will age, too.