From Wiktionary—
So it could be a regional dialectical usage.
Probably not very likely, unless on a kids’ menu, I think. But fairly common on something like a buffet, salad bar, or dessert bar.
And someone mentioned mousse: In the US, that’s a dessert akin to (American) pudding, but aerated into a foamy, slightly firm texture. It’s most often chocolate flavored, but you can encounter mousses of other flavors.
While we’re at it, if you layer together crumbled cake, American pudding, and maybe fruit, you get a dish that’s called either a parfait or a trifle, depending on where you’re from. I think “trifle” might also be a British term?
More recently, “mincemeat” as in “mincemeat pie” is made of chopped fruits and nuts without any animal flesh. However, I think even that usage is almost obsolete.
American pudding sounds to my ears like what we used to sometimes get for dessert in school lunches in Scotland - a custardy concoction known as blancmange. Either plain vanilla, or chocolate.
Blancmange was not considered in any way an aspirational foodstuff, though I kinda liked it. A bit like the greasy pork pie of desserts (I was also fond of greasy pork pies in my youth)
When I was growing up (in Yorkshire), Sunday dinner included Yorkshire pudding for two of the courses. First, as a starter with onion gravy, then as dessert with butter and golden syrup. We didn’t have any with the main course, which is when most other people eat it.
I also grew up in Yorkshire, and that’s partly familiar, partly news. Grandma would call from the kitchen: “Before or with?” and Dad & I would respond (in unison) “Both!”
We’d then get a plate sized one with gravy to start, and one with the roast. I didn’t actually encounter it in a sweet context until I was married and my wife mentioned her mum (from just over the Yorkshire border in Grimsby) doing so with leftovers.
Not quite…“mincemeat” only ever means the fruit version and is used almost exclusively for making “mince pies” (never called “mincemeat pies”, although the relatively rare lidless version would be a “mincemeat tart”). Though this could come from “meat = food” type origins, it’s popularly believed that it historically did contain meat. I’ve certainly encountered the idea of chefs who make a thing of revisiting historic recipes including meat in their mincemeat as a now-unusual quirk.
“Mince”, on the other hand is ground meat, usually beef. Lamb and pork mince are also widely available (chicken less so) but unless specified, “mince” is minced beef.
You can get pies filled with mince, but they’re not “mince pies”. But if in amongst a selection different meat pies, they’d be labelled “mince” (but not “mince pie” - that would contain fruit).
Clear?
Blancmange is supposed to be set through: a sort of milky jelly. American pudding has a viscosity to it, but is more fluid than blancmange.
In the Caribbean part of Colombia, pudín is what Americans call “cake.” This chocolate cake, for example, is just called pudín de chocolate. However, in other parts of Colombia, pudín refers to what Americans call “pudding.” And, of course for those of Mexican origin here is L.A., they say pastel for “cake.”
I’m curious: What do they say in Panama?
South Africa falls somewhere in between the UK and US:
“Pudding” absent context definitely means “dessert” here, same as the UK.
Savoury puddings are definitely not a thing. I had to learn to make Yorkshire pudding myself, and steak-and-kidney is eaten in pie form only.
But baked warm puddings are definitelya thing. And it’s bread and butter pudding here, too. Served with custard, which is a more-or-less runny sauce for puddings, not a pudding itself. Although you can have it by itself. But you’d be weird.
However:
The packet-mix cornstarch-based blancmange-y milk desserts are also called “pudding”, usually in the form “[flavour] pudding”, so “strawberry pudding”, “chocolate pudding”, etc. My favourite is butterscotch pudding. So that usage is like the USA.
Not my experience of school dinners - not that this means you’re wrong!
Yeah, it’s supposed to be set with gelatine, carrageenan or somesuch, but school dinner interpretations of “supposed” weren’t always that accurate! We used to have jelly that you could drink…
Steak and kidney pudding is certainly a variety of pie, but its steamed suetcrust pastry marks it out as different. You’d be able to find them in most supermarkets still, but they’re not as commonplace as regular pies. They’d be amongst the “really…people still by that sort of stuff, do they?” tinned goods.
What sort of philistine would serve Bread & Butter pudding without custard?!
Some more British real puddings:
Spotted Dick (go on, you know you want to)
And others from the Pudding Club
Only “slightly.” 
There’s already custard in bread and butter pudding. Putting extra custard on top would be redundant.
I think there might be a bit of confusion for non-UK posters about pudding
Pudding can be a meal course. It is entirely possible to have a dessert course - which usually does mean consumption of a dessert food but might occasionally be a cheeseboard - sometimes people have the choice between cheeseboard and dessert food - this usually depends upon social context. It’s not unusual to have both a dessert/pudding course and a cheeseboard.
Pudding can also be a food product or recipe.
Some puddings would be savory and would be served in one of the main meal courses, some puddings are sweet and would be served in the pudding course. Plum duff is obviously sweet, whereas dropped-in dumplings are a form of pudding mix but go into the mean course such as a stew, and Yorkshire Puddings are savory and are consumed prior to the main meal - often as a prequel to the roast meat course being served.
The thing about Yorkshires is that the mix is also virtually identical to a pancake mix - pancakes would almost always be considered dessert that is consumed during the pudding course.
For anyone who didn’t go to a British boys boarding or grammar school. This is about school dinners. The sadistic schoolmaster is telling the kid (pink) that he is not allowed any of the (relatively) yummy pudding if he doesn’t finish up the dregs(?) of his main-course meat, possibly kidney (or maybe sliced “beef”) the gristle, burned fat and mystery bits of tube that he’s pushed to the side of the plate.
Meanwhile back to puddings (which are hardly ever cake).
In the US, pancakes are traditionally a breakfast food. You can eat breakfast foods for dinner (some restaurants specialize in this), but you wouldn’t eat pancakes after something else.
And it sounds like blancmange (properly made, not what you’d get in a cheap school cafeteria) would be more like flan? Though flan is relatively rare in the US.