Most likely they didn't. I often produce similar designs while doodling, and I apply Penrose discoveries when I do so.
If they did have such knowledge, and didn't acquire it before Mo's time, they probably stole it from the Hindus in India, just as they stole much of their other so called "achievements" in math. (The pre Mo Arabs were good mathmeticians and astronomers too, which is why many star names are Arabic, such as Alderaan (no longer commonly used, except in Star Wars), Aldebaran and Betelgeuse.)
Alderaan and Aldebaran are two alternative names for one star system (it could be a multiple star star system, but it looks as though it is a single star with the eye and simple telescopes).
Exactly. If anyone associated with such crystalline designs possessed any more significant understanding of the geometry and mathematics associated with it, why did it end there? Why is there no further mathematical tradition associated with this, just a 500+ year lapse??
Yes, these are remarkable and intricate patterns of tiles, beautiful in their own way. No, they do not represent some great leap forward in 15th century mathematics.