I'm not in the US, and product names differ, but I've seen it called BBQ lighting gel, among other things but almost always gel. Small tubes may be found as priming paste/gel for petrol stoves - that's the only time I've seen "paste" used in English for this sort of thing.
If you can find 90% alcohol hand sanitiser, you can use that; it's also gelled alcohol, but much lower concentrations have too much water to be good for burning (most will ignite but soon burn down to a puddle of mainly water).
The advantage of gel over ordinary denatured alcohol is mainly that it doesn't flow and so concentrates the heat where you put it. It may also use a greater proportion of isopropanol, which burns with a more visible flame than ethanol or methanol - and that's good for safety. Otherwise the gel isn't inherently safer, and you still don't want to be adding more to something burning.
You can gel alcohol yourself, but I've never tried it. The key ingredient is calcium acetate, which can be made from eggshells and vinegar. A little water is needed in the mix.