Neither Roku nor Amazon manufacture anything.
Firesticks are made by Foxconn, and always have been - it is Foxconn that is now making them in India in addition to its existing Asian plants.
Roku don't say who makes their equivalent, but their recent annual accounts say:
We outsource the manufacturing of our products to our contract manufacturers and original design manufacturers. All of our products are manufactured in the People’s Republic of China, Southeast Asia, and Brazil.
I believe that Foxconn is one of those manufacturers. So it is true that the same third party manufactures (and continues to manufacture) both, but not that one manufactures the other.
The software of the two is completely different - Fire OS being a fork of Google's Android, Roku based directly on Unix. Roku was originally part of Netflix, but was spun off early in its life.
