Bellman operator and contraction property
The proofs I have seen for contraction of the bellman operator (this page has a really nice run through, explicitly utilize the fact that an infinity norm is being used. This does not necessarily mean that contraction doesn't occur under other norms, but it does suggest one of two possibilities:

The contraction proof is only valid under the infinity norm.
The infinity norm is just the easiest metric to prove the contraction property.

When showing that the Bellman Operator converges to a fixed point it is satisfactory to simply show that it is a contraction, it doesn't matter what sort of contraction it is, so we would typically prove the contraction that is easiest to show.
That being said, intuitively, I would imagine that the question of whether or not this contraction holds under other norms would impact questions surrounding the rate of convergence to the fixed point (ie., how many times we must apply the bellman operator for convergence). After a cursory search I was not able to find any theory explicitly proving or disproving the contraction under other norms. This might be an interesting question to raise in the mathematics stack exchange.