I'm really confused here!
No, you are not! You only (temporarily) mixed up two things, both of with you have already understood separately.
$$\ce{HA + H2O <=> H3O+ + A-}$$
1. Position of the equilibrium
This is described by the equilibrium constant $K$, which is the quotient of the rate constants for the forward and the back reaction: $K = \frac{k_{forward}}{k_{back}}$. For strong acids, of which you know that they completely dissociate, the equilibrium is on the product side.
2. Equimolarity
There's nothing out of nothing! If one molecule of $\ce{HA}$ dissociates is water, there exactly one molecule $\ce{H3O+}$ and exactly one molecule of $\ce{A-}$ generated. Not more, not less.
Q: How could another molecule of $\ce{A-}$ appear out of nothing?
A: It can't! and it doesn't happen - not even in the LEGO universe.
It is just a proton transfer reaction from $\ce{HA}$ to $\ce{H2O}$; the overall number of atoms does not change!
If is however possible that some $\ce{A-}$ may come from a different source, just by adding a salt $\ce{MA}$ of the conjugate base (think sodium acetylsalicylate) to the solution of your acid. jerepierre has mentioned this in his comment! But for the moment, this is another story.