I think "Deportivo" is somekind of Spanish term of sport association like the Italian word "Sportiva" in A.S Roma or S.S Lazio official name. Many sport association in this world that specialized in football sport.
As for the usage of term "Deportivo" instead of "La Coruna" by many people in this world, I think it's already fixed and unchanged for long time. I am not sure when people started to call them "Deportivo" or "Super Depor", as far as I know Deportivo La Coruna was once a club in third division of Spanish La Liga long time ago. There must be many other deportivos in Spain at that time.
Well, it looks little unfair that actually many other deportivos in Spain, like RCD Mallorca or RCD Espanyol. Even small club like Alaves also a deportivo although they're not a "Real Club" (royal club). But, that's reality...there're already too many people in this world that assume there's only one Deportivo in Spain (most of people don't know what the meaning of "Deportivo".
I think this is fine since so far no clubs complaining about that. The real point is the using of "de' in "Real Club Deportivo de La Coruna". It means this club is originated from La Coruna, not a name of the club. Somehow I find it questionable if "Deportivo" is the real name of this club, is it the first sport association in Spain ? I mean it's pretty daring to make a general term as official name....