Are the mini lathe and mini mill spindles the same?

If you are talking about the Haas machines, the answer is no. The spindle for the mini lathe is specific to that machine, if memory serves me, the base spindle for the mini-mill is the same as for the TM-1, TM-2.

Not the same, at least on my mini-lathe and mini-mill (I'm talking Grizzly here). Mill is an R8, little bit different gearing internally, has a drawbard seat on the backside. Lathe is either straight or taper seat on the chuck end.

It's probable the bearings are close enough in size and spacing that you could kludge it up to swap one for the other, but they're not the same part.