Student or Learner
In the case below, should "onion" be plural?
"I'd like a hamberger, without onion/ onions, please."
With things like this, we usually use the singular (mass noun), not the plural.
Q - What does the burger come with?
A - It comes with onion, lettuce and tomato sauce (not onions, lettuce leaves etc)
So "Can I have the hamburger without onion, please?"
Well, 'I had a hot dog with onionS' at a barbecue yesterday. You can't compare lettuce, mayonnaise, ketchup, etc because they're uncountable. My opinion: plural is standard UK English here so a 'hot dog with onions and tomatoes, please!', not 'a hot dog with onion and tomato, please!', which makes sense because you're not having one slice of an onion or tomato (they're countable). I reckon the singular is more of a US usage and if we use it in the UK it's under this all pervasive influence. In short, either, or in UK English.
I agree with your hot dog example, though I would be interested in what you would expect to have been the answer had you asked "What does a hot dog come with?" Would you have expected them to say "With onion" or "With onions"?
Onions (to me) would be entire vegetables! I imagine your hot dog did not come with 2/3/4 whole onions, rather that it came with [some] onion.
Generally, I don't think I'm prone to using AmE very often but, with regard to your example above, I would always ask for "A hot dog with onion and tomato, please" not "A hot dog with onions and tomatoes, please." I wouldn't differentiate between these as being a BrE vs AmE issue.
(I should point out I wouldn't ask for a hot dog or a hamburger at all because I'm vegetarian! )