The characteristic property of all acids and bases is the pH. Their characteristic general reactions are the acid-base reactions. Let's concentrate on this kind of reactions. We don't want to consider here reactions of particular acids and bases (like e.g. oxidation, passivation, complexation).
We have to distinguish between dissociation degree of an acid and acidity (acid strength) of this acid. The degree of dissociation depends on the initial concentration, the acidity not. The acid constant of an acid is a measure for its acidity. An acid is characterized therefore by its acid constant. Acids are grouped into strong acids, medium strong acids and weak acids according to their acid constants.
You are right, higly concentrated strong acids can have lower dissociation degree like weak acids.
But for the behavior of an acid in an acid-base reaction, the reaction velocity is crucial. It depends on the hydrogen ion concentration in the solution. The measure for this concentration is the pH of the solution.
Highly diluted strong acids can have the same pH like less diluted weak acids and therefore the same reaction velocity in acid-base reactions.
The analog holds for bases.