Why would people assume that “Afriki” is Africa?
Because that's how the Greeks (whose influence on Judaism and many other cultures is well documented) spell and pronounce the word Αφρικη until this very day.
Other than the name being similar to the English one, why would one assume that אפריקי in the Gemara has anything to do with Africa? The names are similar, but is there any other reason?
You almost make it sound as if the English invented the word, which simply isn't the case.
The names are similar, but is there any other reason? Example of what I mean, is there anywhere (a Torah commentary, the Gemara itself, etc.) that discusses the place as having animals, features, or the like that make it Africa in particular?
The Greek word literally translates as without cold, a- being the negative particle (as in atheist), and friké meaning cold (as in Romanian frig, for instance). Obviously, Africa is a notoriously warm continent.
The Talmud also speaks of Gehenna (another notoriously warm place) lying beyond the dark mountains, and needing an Egyptian donkey to reach. Obviously, Africa is home to Egyptians in particular, and other dark-skinned people in general, as for the mountains in question, they are described as being incredibly tall, so as to completely block the rays of the sun, a rather transparent allusion to the famous Atlas mountains, whose height was believed by the above-mentioned ancient Greeks to reach heaven, which is why they named them after their legendary homonymous hero, condemned by their supreme god to carry the sky on his shoulders.