To start, Bombay and Mumbai (मुंबई) souding similar is coincidental. Bombay was the Portuguese name, meaning "good bay" Bom Bahia. Bombay is the English interpretation of the Portuguese name. The local name was originally something akin to Mumbā named for the maternal goddess. While I can't say the Portuguese weren't inspired by the old name, the two are not related. The final -ai in Mumbai comes from Marathi, the local language spoken in Mumbai, and means "mother". Mumba-ai -> Mumbai.

The exact etymology of "Africa" — to my knowledge — is not entirely clear. However one explanation is the Latin is basically "place of the Afri/Ifri", as at least one of the hypotheses. Ifrīqiyah (إفريقية) was the name of a particular spot on the north of the continent, to which the Latin etymology wouldn't be a stretch. Today the Arabic version of today is what you see in Afrīqīyā (أفريقيا). A shift from an initial /i/ to /a/ isn't really problematic, anyway.

But here's the most important part:

What accounts for these changes?

Every language has loanword phonology. There are certain rules in your language for which sounds can go together. Certain combination which work in one language may well not be okay in yours, like in English, starting a word with "ng", though ending a word with "ng" is no problem. Or starting one with /ts/, which is why a lot of English speakers say "sunami" for <tsunami>. Certain combinations sound weird in different languages, for the sake of a simple explanation.

So your scepticism about the British mishearing Mumbai as Bombay is correct; They didn't. The Portuguese instead called it Bombay and the British went with that. The Portuguese also called Taiwan Ilha Formosa which survives today as just "Formosa". Obviously that wasn't an issue of mis-hearing "Taiwan".

tl;dr: In most of these sort of cases, it's not that it was misheard. Rather words are adjusted to fit into the phonology of the recipient language (most often) or simply are given another name (slightly less common) and that sometimes sounds very similar to the native name (much less common).

bonus point: Many instances of countries or cities being renamed over time aren't actually cases of being renamed. Bombay>Mumbai certainly was, but Peking>Beijing (北京) wasn't actually a renaming, and neither was Burma>Myanmar (actually Bəma>Myə̃ma ဗမာ>မြန်မာ). Often it's just that names' pronunciations are updated to modern pronunciation from older ones (Beijing being an example) or going from colloquial to literary pronunciations of the same word (Myanmar being an example), or shifting from one dialect's pronunciation to another, such as Rangoon>Yangon (ရန်ကုန်).

edit: typos in romanisation