There's a similarity in the simplicity of ordering, yes.
But that's where the similarity ends.
As an American, even I know that a Full English has a defined set of things included. Sausages, mushrooms, etc: It can't be a Full English without the things that constitute a Full English Breakfast.
But otherwise, the special (or maybe more properly, "the daily special" but nobody ever says it like that) isn't like ordering a Full English at all. The special isn't even a little bit predefined.
The Special could be a club sandwich and potato chips, or meatloaf and mashed potatos, or a burger and fries, or some tacos, or regional fish like Walleye or something. Or whatever.
And the main distinction is that The Special changes daily, and oftentimes there will be different Specials for lunch, breakfast, and dinner.
In American diners, the special is usually written on a sign on the wall and/or maybe outside by the door.
Whatever it is, at whatever time, it's just a reasonably-complete meal that they're selling at a special low price right now that requires minimum effort on everyone's part and that they're prepared to serve a ton of today.
And tomorrow, the special will be something completely different.
A person may or may not be able to order whatever items that comprised yesterday's special, but if they can then the price will be higher today than it was yesterday.
But that doesn't matter, see? Just walk in, order "The Special," and food will show up.
Perhaps the context matters: I am not a fussy eater, and I love street good and cheap eats. I much prefer it to fine dining. But there's a reason for the latter: I'm a lifelong vegetarian. So long as I can get something without flesh in it, I'll eat anything, preferably deep-fried.
But whereas across the British Isles I can ask for a veggie Full English and get something good, I can't imagine walking into a food place and ordering sight-unseen, unless it was a veggie place, and they are typically upmarket and expensive, not the sort of basic joint I happen to prefer.
But reading this and the comments here, it occurs to me we sort of have something comparable: the legendary Full English Breakfast.
(Insert Full Scottish, Full Irish, and Full Ulster as appropriate. I can't remember what it was called but I've had a few Full Welsh as well.)
You go in, and you order a Full English, and that is the extent of your choice. It's generally wonderful.