Groß (in its literal meaning) indeed indicates great spatial extent, not necessary (only) vertically. It is in its generality the German equivalent to "big". A big rock, a big house etc. will probably be wider but also higher than average.
With people though, the meanings of these two words have specialized in opposite ways: In German the literal meaning of groß typically translates to tall (but see the next paragraph for an exception) while in English big translates to German dick. One reason may be that English (but not German) provides a more specific adjective, tall, specifically for height; if one wanted to emphasize height one would use that, so the general big is used for the other spatial dimension. (For some reason in neither language high or hoch are used for persons.)
Another aspect is that big (as opposed to fat) is a mild euphemism; if I'm not mistaken we are lacking an equivalent in German. Dick is pretty blunt, though not as pejorative as fat. This gap can on occasion be filled with groß: If you google "Große Größen" (for clothes) you will not find clothes for tall men: The second hit is "Kleidung in großen Größen – Maximaler Modespaß für Kurven" :-) — obviously one wanted to avoid calling the customers dick and emphasized the attractive part of body fat, namely the curves, as one would in English.
As an aside, this is a common error for Germans who learn English. Big and groß are practically equivalent except when it comes to the dimensions of human's bodies.