Is adding butter to caramel sauce essential?

Yes, you can make a caramel sauce without butter, but you have to add either water or... something else.

If you use water, you'll get a pure caramel syrup that looks like honey. Personally, this is what I'd recommend because it's versatile on it's own, and can be repurposed into a different caramel sauce if you add another liquid, and patiently cook the mixture back to the proper temperature.

In order to make a pure caramel syrup: Cook your sugar up to the desired temperature in a very large pot. Then, remove the syrup from the heat (I usually move to the sink to be safe), stand back, and using a towel or oven mitt to protect your hand, pour water into the pot. There will be lots of steam and bubbling, so please be careful. Then return the caramelized sugar and water to the heat, stir to dissolve the solid bits of caramel, and continue to cook until you've reached the consistency you desire. For a drip/glaze, you'll want to cook it to between 225 and 235 F, I'd think. It would be best to keep a cool plate to the side so you can drip a little of your caramel syrup at a time and judge its thickness as you go along.

Keep in mind this will be a syrup by itself, so it will drip... and keep dripping. It will not set up, unless you cook it to the point where it's chewy/tough like @rumtscho warns. However, an easy workaround is to make a piping gel, either with gelatin or cooked starch using your caramel syrup in place of corn syrup or sugar.

If you want an opaque syrup that has a more complex or creamy flavor, you'll need to use a dairy product instead of water. Butter is a go-to because it lends the most richness with the least amount of steam, and least dulce de leche/cooked milk flavor, and accordingly the least risk of curdling. Heavy cream is also a go-to because it's a very stable emulsion, which will help prevent any greasiness from separation, is still unlikely to curdle, and doesn't produce a lot of steam. Some of us really enjoy the tart dulce de leche flavor as part of a caramel as well.

You can use lower-fat dairy products like half and half, or even whole milk. I can't tell you if 2% milk, skim milk, or non-dairy options will work, but if they can handle being boiled without curdling, then they'll probably work. I just can't promise it. The lower fat and higher protein your liquid has, however, the more danger you'll have of curdling. This danger is exacerbated by the high temperatures, and by the darkness of the caramel. The darker the caramel, the higher the acidity. So, if you use a lower-fat liquid (other than water, obviously), I would recommend letting the caramel syrup cool down significantly before adding the liquid. That will be more time-consuming, but will hopefully prevent curdling.

Regardless of the substance you add, the basic procedure remains the same: Cook sugar in a big pot until it's the right temperature/color for your needs. Then, (allow the molten sugar to cool down if adding a low-fat liquid, or else take measures to protect yourself from the steam, and) add the butter or liquid. Return to the heat, stirring to re-melt and combine. Continue stirring and heating until it reaches a good temperature/consistency for your application. Consider using it to make a piping gel if you need more stability.

Also, in case it's relevant-- I've tried making caramel using coconut oil, but that did not work. It totally separated. Adding heavy cream managed to emulsify it, but it still had a distinctly oily taste and feel to it, which I did not enjoy. I did not test using coconut milk or coconut cream, but considering that those tend to separate in the can, I'm skeptical that they would work well either.

If you don't add the butter, it won't be a sauce, it will be pure caramel. Caramel is not a sauce, and is not suited as a cake glaze - it will stick more to your teeth than to the cake if soft, or be an uncuttable brittle plate if hard. If you need a sauce, follow the recipe for a sauce.

It is possible to make caramel sauce without butter, but you'll need something instead, and that's likely to be another fat.

For example here's one that uses lots of cream.

Vegan recipes of course do without butter. Some use a butter substitute but many seem to use coconut oil, coconut cream, and/or coconut milk.

You may need to adjust the proportions to get the thickness right for pouring over a cake