They are actually made quite similarly, if you are talking about Texas red and the stew version of goulash (not the soup, which is what goes by gulyás in Hungary.) The meat in Texas chili is just cut up a bit finer, and there is the additional spice of cumin.
The carrot and celery in my chili Are both minced to a smooth paste. They add a depth to the flavor and bulk to the sauce. Tomato goes a long way flavor wise so I only add the whole pealed one and no sauce or paste. To me they just make the chili richer.
Interestingly, not only is Texas Red not much like what I think of as chili, but that porkolt isn’t much like what I think of as goulash, either. For starters, the goulash I know has a lot of potatoes and onions, and I’m pretty sure several other vegetables too (it’s been a long time since I’ve had it).
That porkolt has a lot of onions in it, but they’re probably just cooked down so you can’t see them in the photo. The ratio varies, but I typically go for 2:1 by weight of meat : onions. For example, here my picture of a completely to-spec Hungarian porkolt made with that ratio of beef to onions. Meanwhile, this is what goulash (the soupy version) looks like. The “stewier” version is made with just meat, paprika, and onions for the most part. If there’s potatoes, they’re cooked separately. The soupier version will contain ingredients like potatoes, carrots, parsnips, celeriac, etc., but is much more a soup than anything else. (And even the carrots, parsnips, celeriac, etc., are optional. The main requirement is for it to be a soupy version of porkolt with potatoes. In fact, the recipes are almost the same, with the main difference being the amount of liquid added).
As for the Texas red, that actually is a lot more coarse than what I typically think of as Texas red, too. Something more like this is what I think of. Just google image search “texas chili” to get an idea.
