Evidently they do. The M60 tank had a 105mm gun (so did the original M1; it didn't get the 120mm until the A1 version). There's also 105mm recoilless rifles and 105mm howitzers but none of them fire the same round. The 106mm recoilless rifle is really 105mm; it's referred to as 106mm in order to make it clear that ammunition from an earlier 105mm weapon is
not compatible with it.
As for the M1A1, the problem with the 120mm round is that no dedicated high explosive 120mm round exists. Several rounds might suffice, such as the M830 (HEAT) or the M980 (obstacle reduction), but both are designed to hit things like light armor or concrete, not snow and therefore you might get undesireable results. The M60 actually retained certain minor advantages over the M1 series; one of them was the greater variety of ammunition for the 105mm gun, which ironically was not a problem on the earliest M1s. HE rounds are also cheaper than the specialized rounds used by the 120mm, and the diesel engine of the M60 is more economical to operate than the M1 turbine engine. The M60 is also about 17 tons lighter, better for the roads.
This round is designed to combine the 3 modes of the M830 plus the M1028 into a single round, and might have been suitable for avalanche control, but I don't know the status of the program, and there'd still be the problems of the fuel and extreme weight of the M1A1.