For fonts every big company does its own thing too. San Francisco is an Apple font you won't find outside their ecosystem. Arial, Courier New, Comic Sans, etc are Microsoft fonts (or popularized by being shipped on Windows 3.1). Roboto is a Google font and used by all their products.
It just happens that fonts are mostly handeled by operating systems providers while emoticons are mostly handeled by chat system providers. That has historical reasons since beautifully rendered emoticons were used as a selling point long before emoticons were added to fonts in the West (Japanese code pages did have emojis, so things might have developed different if chat was dominated by Japanese companies instead of American ones). Meanwhile providing custom fonts wasn't feasible for websites for most of the internets existance, so that hasn't caught on yet as something web-based companies like twitter do.
It just happens that fonts are mostly handeled by operating systems providers while emoticons are mostly handeled by chat system providers. That has historical reasons since beautifully rendered emoticons were used as a selling point long before emoticons were added to fonts in the West (Japanese code pages did have emojis, so things might have developed different if chat was dominated by Japanese companies instead of American ones). Meanwhile providing custom fonts wasn't feasible for websites for most of the internets existance, so that hasn't caught on yet as something web-based companies like twitter do.