I think the problem is:
Attack is both a generic thing (I attack somebody), and a defined term.
An Attack Action is a standard action that involves attacking someone.
Use of Manyshot is a standard action that involves attacking someone. But Manyshot is not an Attack Action.
Why? Well, there is some room for ambiguity, but it boils down to 'just because two things describe the same event, doesn't mean they are the same.' That is, an Attack Action is not _any_ action that involves attacking someone. There are Full Attack Actions, after all, and special actions that feats describe.
But this gets murky. Is _any_ feat that attacks with a full action considered a Full Attack Action? By what we call around here a 'German' reading of the rules, not unless it states so.
Shot on the Run says 'when using the attack action with a ranged weapon.' Note it says '-the- attack action'. if it said 'an attack action', there would be at least some indication that they are generally referring to any attack. Though they would then leave it open to using full attack (which is _an_ attack action) and so forth.
I'm contemplating house ruling Manyshot to be used with Shot on the Run, for what it's worth. I just don't think the rules as written support it.
In addition, Manyshot clearly only applies to arrows. Perhaps a balance issue, but it's pretty straightforward.