According to Judd's in-game tip, abilities do stack, but are affected by diminishing returns, which means that using two or three of the same ability will result in less than two or three times the effect of having one.
It's important to recognize the difference between primary and secondary abilities. Primary abilities appear as a big circle on each piece of equipment, which secondary abilities appear as smaller circle. There can be up to one primary and three secondary abilities per piece of equipment. According to the Splatoon Wikia:
A secondary ability's value is around 1/3 that of the same primary ability.
Stackable abilities affect your stats multiplicatively, so each one that you wear will improve its respective stat by a certain percent. For example:
The first Damage Up Primary applied gives a 9% damage boost, but 2 give ~8.25% each, and 3 give ~7.45% each.
The Inkipedia and Splatoon Wikia ability pages currently have conflicting information regarding the effects of individual abilities. You should view each to get an idea of how much of a boost each ability provides, but keep in mind that research is still ongoing.
Also, note that Tenacity cannot stack because it is only available as a headgear primary ability. All non-stackable abilities are only available as a primary ability on one type of equipment, so you never have to worry about wearing two non-stacking ones.
The 13 stackable abilities are:
- Damage Up
- Defense Up
- Ink Saver (Main)
- Ink Saver (Sub)
- Ink Recovery Up
- Special Charge Up
- Special Duration Up
- Special Saver
- Run Speed Up
- Swim Speed Up
- Quick Respawn
- Bomb Range Up
- Quick Super Jump