There is a dojo...? here that has only one word on its sign: "Karate" The main art taught there, the art that the owner is ranked in is TKD. They teach other arts there as well... a style of Kung Fu, Aikido and some sort of MMA class. What they do not teach there is Karate. I asked him why the sign said Karate, when they have never taught that and his response was "Everyone knows what Karate is, not many people know what TKD is." And this place is a commercial dojo, part of one of the larger organizations of TKD schools in the area.
Asking the instructor what the art is that he is teaching is the way to go. It may be his own blend of Karate and TKD and a few other things. As long as he is up front about that, and you enjoy the classes... nothing wrong there at all.
i am surprised the OP has not asked the owner, instructors, or fellow students. That is something that is most often asked before signing up and paying money. But I accept the OP may be a bit shy.
I agree with others who have said if what they are being taught are viable self defense moves and being taught in when and how to use them, and the OP enjoys them, stay with it.
In the 60's and 70's most TKD schools put karate sign out front, because people had hear of karate and few people had hear of TKD.
in the 80's because of the Karate Kid most people call any MAs karate.
True ,TKD is the #1 most commonly practiced martial art in the world, but most people don't know MA and call every MA karate.
Now that I am training in karate, I think of tradional TKD as Korean Karate.
I know those who study TKD will think this is blasphem, I would have,
but when talking to normal people about MA it is so much easier.
When I studied TKD in the early mid-sixties, we knew what we were learning, but admitted when we talked to outsiders we usually told them we studied TKD, and waited for them to ask if they didn't know what that art was; that it was sort of like a Korean karate, but different. TKD just wasn't a well known art at that time.