Ketchup only being made from tomatoes is a comparatively recent thing.
In the UK in the 1500s or so the most common condiment was mushroom ketchup, which you pretty much have to make from scratch these days and that's a bit of a pity. You cook the mushrooms off with mace, nutmeg, ginger, pepper, maybe some tamarind (that's the stuff that gives Worcester sauce its sour flavour - it's got the same thing in it as sorrel which I guess you could also use but that's better for like a pesto), brown sugar, and some red wine, and just keep going until it reduces down into a thick sticky paste.
It's quite a strong flavour, probably a bit full on for the average American palate, although I know people over there who make it and apparently enjoy it.
Tomato ketchup is like a cross between a pickle and a jam, where (at least in theory) the sugar and vinegar act as preservatives and the pectin from the tomatoes makes it set up into a sticky jelly.
People also made walnut ketchup, which is difficult to do where I am because you need green unripe walnuts and there aren't a lot of walnut trees here. If you live near a walnut grower see if you can get a bag of unripe ones and have a go at it. The unripe walnut juice will stain the hell out of everything it comes near - it'll stain Pyrex, ffs - but it's good stuff that goes nicely with a bit of venison.
They have a spicy ketchup product that I liked, tho it has disappeared from my local stores. At the time I first noticed it at the store it was the hottest one that also matched the thick consistency I expect from ketchup.
There are definitely better things out there.