Analog computers don't have infinite precision due to the presence of noise, so digital computers can emulate that with high-enough precision arithmetic.
Yes but there are equations (like stiff differential equations) that are extremely difficult to solve accurately with a digital computer but which are trivial on an analog computer.
Simulation is about mimicking another device or system. Emulation is about setting up a system that is logically indistinguishable from another irrespective of its implementation substrate and details thereof.
A thing is successfully ‘emulated’ when it is logically impossible to distinguish the difference between the system and its emulated counterpart.
Since, as you said in a sister reply, even one analog computer might be slightly different from another analog computer, and thus unable to emulate it, if you had the outputs of two different computers, one analog and one digital one simulating it to a high precision (higher than the noise of the analog one) how could you distinguish which was digital and which was analog?
If you can't, then this is a meaningless semantic discussion. The digital computer can emulate the analog one as well as any other analog computer can.
The point is that discrete computers can exactly and trivially emulate each other. The inability to emulate an analog computer by a digital or analog computer kind of is the whole point.