For anyone having the problem as well, the following insights from support chat:
Emojis currently only supported for Windows platform Windows 8 and up (8.0, 8.1, 10) with Internet Explorer or Firefox.
Other browsers and operating system may have different results, I've tested so far on boonex.club domain:
Windows 7 with Opera browser: no emojis at all
Windows 7 with Firefox, Chrome, Internet Explorer: emojis in shoutbox appear black and white and very small
Windows 8.1 with Firefox and Internet Explorer: emojis appear as they show in drop down list but colors may vary (like the blue shirt in list appears as yellow one in shoutbox)
Tests with Linux (Mint) and different browsers just show broken characters as shown above.
Trying to test Windows 10 later as well.
In general not very great solution at all if a good percentage of web users are unable to see the emojis in shoutbox and messenger. Even if we just talk about Win7 it has over 50% market share still according to studies. Maybe some issue to investigate @AlexT ?