tl;dr: You can’t do it without actually making it an emoji, or using a custom font that has the character.
Emoji are text. Unicode is the global standard for text, and especially for large character sets. This is the strength of emoji and what makes them work as well as they do. (Not perfectly, but as well as reasonably possible.)
Unicode (including the emoji parts of the standard) is exactly what enables emoji to be copy/pasted as text. It is a vital part of the tech behind emoji. Emoji have standard Unicode codepoints, and an understanding of those is shared by essentially all the world’s computers and digital devices… and that is why emoji work across nearly all different devices and technology platforms.
Just like any other text, you need a font that supports the particular characters in question; in this case, the particular emoji characters. Luckily operating systems (and some apps) come with emoji fonts for this purpose, and those fonts get regularly updated, so when you update your OS (or app) you get a newer font with the latest emoji included.
But let’s take all the bits of the question one at a time and break it down for you.
(This post started as one of my Quora answers, years ago. Since revised and updated.)
First, what do you consider “normal text”? You must be part of the minority 15% of the world’s population that has English as their first language, so even ASCII is more-or-less sufficient for you. But for the other 85%, there is no single obvious standard… other than Unicode. And in fact, Unicode is “normal text”: starting in ~ 2011, more than half of the world’s 10 million most-used web pages were already encoded as Unicode. By 2023 it was about 98%. So, I think it is safe to say that the encoding used by 98% of all web pages (and a similar proportion of desktop applications and operating systems) is normal text, pretty much by definition.
Let’s temporarily ignore that, and still talk about ways of communicating “something like an emoji” without Unicode.
What counts as “something like an emoji”? You can have an image in HTML/CSS that has “alt text” with a description. That text is normal text, and the full text includes the URL link to the image. That doesn’t require Unicode, something that supports even one language is sufficient (for just that one language!). That does not seem much like an emoji to me, but fine, the stuff you paste is indeed treated as text. However, the image part of that is not text, and will not come along when you copy/paste text alone. So… maybe that part is a “fail”?
Another obvious answer would be… you can invent your own competing standard to Unicode! It would have to work pretty much the same, including the idea that all operating systems and software vendors would need to support it. So you would have to convince everyone that it was worth doing so. That’s a process that took decades for Unicode, even though all the biggest players in operating systems and software were backing Unicode, and there wasn’t a viable alternative!
So if you want to compete with Unicode… well, good luck with that. It was long and slow getting everything to support Unicode, even though the alternative was badly broken; even if a slightly better alternative existed, it would not be worth the immense effort to switch or support it, because the additional benefits (if any) would not be worth the effort. Unicode has its flaws (ahem, Han unification, ahem), but I don’t expect to see any real contender to Unicode in my lifetime.
OK, there is a slightly more sane answer, but it is not terribly reasonable, for most people. You could design/engineer a custom font that uses an arbitrary non-standard code (could even be ASCII!) or a ligature (so you can spell out the word) and then create a web site that hosts that font, and then you can input your own text on that web page, including your custom emoji-like character. And it will show up just fine. Until somebody tries to copy and paste it… anywhere else. At which point it will be broken. So it doesn’t really meet that important criterion.