Emoji are almost exclusively used embedded in regular text. Emoji work similar to our regular alphabet, only that they are typically used to express emotions and abstract concepts that are harder to express in regular words. Since we have regular text and emojis used in the same context (and often as part of the same sentence) it makes sense for unicode to support both.
This is fundamentally different from icon fonts. twitter icon or mailbox icon are not meant to be used inside of text, so they have no business in fonts.
This is fundamentally different from icon fonts. twitter icon or mailbox icon are not meant to be used inside of text, so they have no business in fonts.