Midoriya Izuku and Bakugo Katsuki have known each other since they were little kids, so of course, they have pet names for each other. Midoriya refers to Bakugo as "Kacchan." In turn, Bakugo refers to Midoriya as "Deku." They each seem to be comfortable with these names, as they both use them and allow them to be used throughout My Hero Academia.
Despite how often these pet names are used, they aren't necessarily terms of endearment. Kacchan is sort of cutesy for a teenager to go by. Meanwhile, Deku is meant a straight insult. Midoriya can take things like this in stride, but Bakugo seems like he would have snapped at the green-haired pain in his side at least once. In the manga, he responded to one of Midoriya's compliments by biting the word bubble, so he should definitely take issue with something that can be perceived as a slight against him. Figuring out Bakugo's complacency in this matter requires an understanding of the boys' nicknames and relationships.
The first thing to understand is the actual meaning behind Kacchan. It's a portmanteau of Bakugo's first name Katsuki and the "chan" honorific. This is normally used for small children, cute girls and close friends. It can also be used for older boys, although it's considered a bit odd. Midoriya probably uses this honorific because he sees Bakugo as a close friend. However, since Bakugo's idea of camaraderie isn't shared with Midoriya, he may think of the "chan" in the child-like sense and perceive it as a passive insult.
Whatever Midoriya may or may not insinuate by calling Bakugo Kacchan, Bakugo has come up with a much worse name for his "friend." The term "Deku" is an alternate reading of Izuku (出久). Deku can also refer to a wooden figure or puppet (木偶), which is commonly used as an insult for those who can't accomplish anything. The Funimation dub takes the clever route of implying that Deku is an abbreviation of "defenseless Izuku." Whatever the meaning, Bakugo clearly means to tear Midoriya down with his name-calling.
Luckily, Midoriya was eventually able to change the meaning behind his nickname to one of empowerment. As Uraraka Ochako points out, Deku sounds similar to the verb meaning "to be able to do" (できる dekiru). Midoriya took this a step further by making Deku (デク) his Pro Hero name. Bakugo, unfortunately, cannot do the same thing with a cutesy nickname like Kacchan.
While Bakugo can't do anything to make the name Kacchan sound better, he might feel he doesn't have to. For the longest time, he's allowed Midoriya to call him by this nickname because nothing he did mattered. Since Bakugo always saw Midoriya as somebody beneath him, he never had to worry about what he thought of him. It wasn't until the darn nerd enrolled in UA's Hero Program and started outperforming him that the boys' power dynamic was put into question.
As for why Bakugo allows Midoriya to continue calling him Kacchan, he might see it as too late to change things. Midoriya has been using this name since the two of them were at least four years old, maybe even longer. Again, this was fine as long as their power dynamic didn't change, which it didn't until they entered high school. It would be awkward to raise a fuss over a nickname that's been used for over a decade.
In all likelihood, the name Kacchan will stick for a long time. Even if Bakugo takes exception to it, making a big deal about the name now wouldn't look good on his part. Fortunately, the way these two boys act around each other has started to look more like an actual friendship than a bully-victim dynamic. It's entirely possible that Bakugo has started using the name Deku for its positive connotations rather than its negative ones, although there's no real way of knowing that (and that's probably how he wants to keep it). This sort of change can't be reflected in Kacchan, but that's okay since it was always meant in the nicest way possible anyway. As long as Kacchan and Deku remain friends, they will continue to call each other by their respective nicknames.
