Do you consider Tang Soo Do an extension of the Karate family?

Yes it is. It came from karate. It retain much of what was passed on to it from karate. When other Korean arts decided to change their kata in order to be different than there once oppressors the people that trained in tang soo do left things as it were.

Even though they leaned there martial arts from Japanese they retained the Okinawan names instead of the Japanese names for their kata.

Edit:

I will change my answer a little. I don't know if extension is the correct term.. it may be semantic. But I was taught too that all is contextual. The style of tang soo do that still uses all of the same kata and names of those kata is nothing more than a form of karate that was taught in Korea and to the Koreans and given the name tang soo do. If they have removed the names and forms/kata then it is something completely different.

An "extension" suggests that it relies upon that which it extends.

That is not the case, in any style of TSD.

If every Karate student, instructor, school, and organization were to somehow disappear off the face of the earth, TSD students would still go to their dojangs and practice; instructors would still be certified; dan certificates would still be issued.

So no, it is not an extension. Karate may have significant influence on the style - of which there are two main flavors, and you identified one "Tang Soo Do Karate", which is what was formerly called "Korean Karate" as a means for instructors in the day to distinguish from "Japanese Karate". The other type of Tang Soo Do is wholly Korean, and has eradicated any Japanese influences, including their forms, technique names, teaching style, uniforms (!), and other elements of Japanese practice.

Tang Soo Do is basically "Taekwondo" by a different name, with slightly different rules. That's why a black belt in Tang Soo Do can be given an "honorary black belt" in Taekwondo without having to start at white belt level and graduate all the way up to black belt level. Taekwondo is the Korean variant of Japanese Shotokan karate. The Koreans were occupied by the Japanese for 400 years, and they resented it, so they renamed the martial arts that the Japanese brought them.

I actually consider it Karate still. It is Okinawan karate with a fresh coat of Korean paint. It is actually not different enough to truly be anything other than Korean karate whether they and others wish to believe that or not. TKD has some argument for being their own since they implement Taekkyeon and other methods of how they do their movements. That is just my opinion on the matter.

No more then I consider karate an extension of kung fu.

Its only part of the major influence.

Mostly, yes. Technique-wise, it has more in common with Shotokan than any native Korean systems, down to the forms.

Yes

I would consider it a VARIANT not an extension of.

I think it is an orange flavored drink.