Well in the days past, that little alligator was sewn onto the best shirts made. it was the start of market logo's.
I still have shirts from the 70's with that little gator sewn on there, they were pure egyptian long fiber cotton, that stayed soft forever, never got boardy feeling like some cotton made stuff. The correct Izod Lacoste shirts used to have only french wording inside the labels. anything with english in it was either Paste or stuff that caused the original company to seek bankruptcy, I forget who the american importer was, but he finagled some sort of a ruling more or less saying that he owned the brand in the USA< and he drove the product into the ground, nearly killed the French mother company and ended up with the IZOD name being owned by some "value added marketing consortium"
The original is still good stuff. now they have miracle fabrics and stuff but the good shirts are still made in cotton and will last for ever.