Force emoji to display as it was sent
Emoji's are special "character" codes. Really, just numbers that all* software agrees that a particular number means a particular emoji.

Custom "fonts" change how those are finally displayed to the user—how exactly each emoji looks. But under the hood, it is just sending/receiving a number.

So when another user gets a message including an emoji number, it loads up the fonts that user's device thinks that emoji number matches.

No other data is transmitted besides that number.

Therefore no, there is no way (through standard SMS/MMS/IM protocols) to supply which custom "font" you used.

The only way for this to be possible if both sender and receiver agree to use a special communication tool that adds the extra "font" information to the message. 

I'm not aware of any specifically do this. Others may correct me on this. But either way, there isn't a standard communication protocol to share an "emoji" font choice.

** Well, most software, agrees on emoji codes. There are some companies that disagree in some cases. This is a known issue with emojis. There is a standardization committee, but again, not everyone agrees with them. Keep hope though; emojis are more standard these days then they were in the past. And maybe the standard will extend to allow for communicating custom emojis (or whole custom emoji fonts) over the protocols? Hopefully, one day in the future, we will have a utopia of emoji agreement from all!