Are all strong acids the same strength? Since the six strong acids dissociate completely into their ions, and the reaction goes to completion, does this mean that acids such as $\ce{HI}$ and $\ce{HCl}$ have the same acidic strength?

I know that 0.10 M $\ce{HI_{(aq)}}$ and 0.10 M $\ce{HCl_{(aq)}}$ both contain 0.10 M $\ce{H3O+_{(aq)}}$, which leads me to believe they are equally strong acids. Is this correct?
There's something called the "solvent leveling effect". In short, there's a lowest pKa in a particular solvent, based on the basicity of the conjugate base.

In water, you're limited by $\ce{OH-}$. So given the same concentration of $\ce{HI}$ and $\ce{HCl}$ in water, they will indeed have the same pH.

In other solvents, you may have a difference between acids, depending on the pKa.