So think of how conservative it is. Someone older, and more conservative could pull of a double breasted suit jacket where someone younger would look like they are wearing a costume. Where something like a single button jacket would is a bit flashier and would fit someone who is much more out going and confident. There are a lot of variations on the same thing. I can just think of styles that run the scales of traditional vs modern and flashy vs conservative, but with those 2 axises you can do a lot to find something to really accent someones personality and style.

Buttons have to deal with the level of formality.

Single buttons have their roots in the single-breasted dinner jacket or tuxedo. If you want your jacket to have an increased level of formality, you'd go with a single button.

Two-button suits are more standard, common for business and every-day suiting. Formal enough for the majority of situations when you'd wear a suit, but not quite "after six" level of formality for an evening event or dinner.

Three-button jackets should be left to casual suits or odd-jackets. Adding more buttons on the single-breasted suit adds an increased level of informality. Works best on casual, seasonal fabrics like seersucker, linen, tweed, corduroy, etc.

Double-breasted jackets have their roots in the military and are often paired with peak lapels, which are more formal than the notched lapels of most suits. So, a double-breasted suit would be definitely on the more formal end.

So, I wouldn't say that buttons are about "personality" as much as it is the situation in which you'll be wearing the suit.