Yes, these “stack,” in the sense that both apply to the same summon at the same time. On some level, this is a little bit of “nothing says they don’t” here, because Augment Summoning and Ashbound are doing different things to a summoned creature. The closest we get to an actual rule is the general statement that
Spells or magical effects usually work as described, no matter how many other spells or magical effects happen to be operating in the same area or on the same recipient. Except in special cases, a spell does not affect the way another spell operates.
Augment Summoning and Ashbound aren’t spells, obviously, but the concept is the same: they both do what they say they do, unless one or the other says they conflict, or some other rule says there’s a problem. No other rule does.
There are general rules, not specific to magic, for stacking, but they don’t apply here. Strictly speaking, in this sense, Augment Summoning and Ashbound don’t stack—because they don’t have to. The two feats apply to different things. Augment Summoning adds bonuses to the creature’s Strength and Constitution scores, while Ashbound adds a bonus to the creature’s attack rolls. Since those are different stats, the stacking rules don’t really come into play—you’re never looking at multiple bonuses added to the same number from these two feats, so you don’t even have to worry about how the stacking effects rule applies.
Augment Summoning’s bonus to Strength may indirectly increase the creature’s attack roll (for melee attacks that don’t use Weapon Finesse), but from the attack roll’s “perspective,” it still is just getting a bonus from the creature’s Strength—that bonus just happens to be bigger if the summoner has Augment Summoning. But the size of the Strength bonus doesn’t matter to Ashbound: Ashbound is going to add its bonus no matter what size that bonus from Strength is.
And anyway, to be really thorough, even if we imagine that Augment Summoning and Ashbound did both add a bonus to the same stat, and the rule for stacking effects did apply, by that rule they would stack because Augment Summoning uses enhancement bonuses while Ashbound uses luck bonuses:
In most cases, modifiers to a given check or roll stack (combine for a cumulative effect) if they come from different sources and have different types (or no type at all)
Augment Summoning and Ashbound are different sources, and enhancement and luck are different types. Again, this does not matter in this case, but in the hypothetical case where it did, they would still stack.
Specific example: Wolf summoned with summon nature’s ally I
If they do a summoned wolf would get +5 to their attack not counting their base (+3 from Ashbound, +2 from Augment Summoning), correct?
This is correct:
Augment Summoning’s +4 enhancement bonus increases its Strength from 13 to 17, increasing its Strength bonus from +1 to +3 (net increase of +2). The Strength bonus is added as part of its bite’s attack bonus, so that increases (due to Augment Summoning) from +3 to +5 (same net +2).
The Strength bonus also increases the wolf’s grapple from +2 to +4, and could come up for things that require plain Strength ability checks.
Augment Summoning also increases the wolf’s Constitution to 19, giving them a Fortitude save bonus of +7, and hp of 2d8+8 (average 17 hp, +4 above that of a usual wolf). Like Strength, Constitution ability checks could come up as well.
Ashbound adds a +3 luck bonus directly to the attack bonus. By itself, this would increase a wolf’s bite attack bonus from +3 to +6, but when you have both Ashbound and Augment Summoning, they increase that +3 to +8, for a net of +5 as you calculated, instead.
- Ashbound does not increase anything else, such as grapple bonus or Strength checks.