I stand corrected, the OS is the arbiter, plus any additions that people apply.

like browsers
IE doesn't like the newer emojis and those it does it defaults to B&W simplified versions (eg 3 wise monkeys)
Firefox 34.05 as IE
Chrome 40.0.2214.111 similar - no modern emojis
Chrome 53.0.2785.116 similar - no modern emojis - Joe Offer's monkeys in B&W
Opera 12.16 similar but no 3 wise monkeys
Opera 28.0 as 12.16
Safari - who cares? The only one I can install is so old I would only use it for verifying really tricky JavaScript wot I wrote.

So for the average user, basically, your browser is the key.