Drill 1 will only be applied once to the same unit. The game only applies the specific promotion specified within the XML-code for the Alhambra wonder. So, no, Alhambra won't give you Drill 2 when you create a Minuteman in the city with Alhambra.
If you are playing as America and have constructed Alhambra you still get the Drill 1 promotion applied to Mounted, Armor, and not-Minuteman Gunpowder units coming as a direct effect of Alhamra in that city, so I wouldn't consider it as having been made 'obsolete'. While Drill 1 is also applied to Melee units, by the time you can build Minuteman, those Melee units cannot be contructed anymore).
It is however a consideration as to whether you will get enough benefit from the free Castle and the other units that can be given the free Drill 1 to make putting the effort into constructing Alhambra worthwhile in terms of Rerturn On Investment.
If your playstyle is not to construct many Mounted units, then for that period of the game when Minutemen can be built Alhambra becomes less valuable, but the value returns after you discover Rifling and cannot build Minutemen any more.
Note: I am using the 'technical' definitions of Melee, Mounted, Gunpowder, and Armored units as it is used in the game's code, which is not the same thing as is often used in 'idiomatic' terms in discussion. Minutemen belong to the UNITCOMBAT_GUN designation, and not the UNITCOMBAT_MELEE designation.