While the Gundam series technically falls more into the sci-fi genre, the argument can be made that it also falls into horror. While it isn't as terrifying as other series that are more overtly scary, it does have an atmospheric horror that's hard to miss.
According to literaryterms.net, horror can be defined as a genre of fiction that "creates a feeling of fear, dread, repulsion, and terror..." The website also states that horror can be broken down into three categories: Gothic horror, supernatural horror, and non-supernatural horror. It's that third category that would fit the type of horror Gundam falls into. It focuses more on the possibility that what is happening in the story could be replicated in reality.
The fights using the giant mech are akin to when a kaiju attacks a city in a monster movie. The only difference is one is a mindless beast, and the other is being controlled by a human being. Many of the fights are a result of some type of war, so it's unlikely that unless the opposing side had intel ahead of time, the citizens of these cities wouldn't have enough time to evacuate and would ultimately be crushed either by the mechs or the falling debris if they didn't make it out in time.
What about the fights that happen in space? Well, a lot can go wrong there, too. If a Gundam's cockpit isn't sealed correctly or is damaged in some way, the pilot would suffer a horrible death wherein the rapid decompression of air could cause their lungs to rupture. It could also cause gas bubbles to form in the blood vessels, causing blockages. There is also the possibility that a ship or a mech could accidentally fly or be knocked into a black hole. Black holes are not visible to the naked eye, though you can figure out they're there by looking at the way they affect gravity on objects around them. If someone were to be sucked into a black hole, they would be torn apart because of the intense gravitational pull of the event horizon.
Let's also look at the fact that the Gundams run on nuclear power. We don't have to imagine what would happen if something went wrong and the radiation leaked out -- we have situations like Chernobyl and Fukushima to give us an idea of what could happen. But what isn't usually talked about in these situations is what happens to a person should they suffer from radiation sickness. Depending on the amount of exposure, the symptoms can be horrifying. White blood cells would begin to die and the skin would burn to the point that it would fall off -- and that's not even all of the symptoms.
The horror of this comes from the fact that it wouldn't just be limited to the pilots should the leak occur inside the maintenance area. Many times, they're kept within mobile vehicles like ships, so everyone on board would be at risk. Not only this but should they be in a city when this happens, the people within that city would also be at risk. Should the crew not realize there's a leak in time, they could spread the radiation to multiple cities.
Perhaps the most terrifying aspect of the series, however, is the fact that humans are to blame for everything. Every season revolves around war in some aspect. The horror comes from the normality of conflict within these universes, and the disregard for life. Each side uses these giant weapons of mass destruction to try and win. Sometimes, it's justified for one side to retaliate or to try and gain their freedom from a cruel government. But regardless of the justification, the fact of the matter is that the origin is rooted in human hatred and cruelty. Many of these people are mass murderers. The only difference between them and a serial killer is one kills during war, and the other during peace.
Much of the horror of the series comes from atmospheric tension of who will live, who will die, and who will win the war. People get attached to certain characters, and to watch them die rips their hearts to shreds. The horror also comes from what could happen to innocent people if one thing goes wrong, even if it happens outside of battle. It doesn't matter if the war is in space or on the ground, there are many ways something could go wrong. There is nothing that these people can do to protect themselves either. Even if their side wins, there's the possibility that the destroyed mechs are leaking radiation into the city or into the vehicles used by both sides to house the Gundams. So, yes... Gundam is a pretty horrifying franchise when you really think about it.