How come hundreds of cultures around the world were able to craft and use the bow and arrow, with pretty much the same design, while having no contact with each other?

Archery was invented around 30,000 years ago.

Back then North- and South America weren't inhabited.

Inventions like bow and arrow tend to spread slowly, if only there is physical connection. Using archery probably spreaded all over inhabited world in over couple thousand years. When the Americas were inhabited (around 14,000 years ago), people brought bows and arrows with them.