Almost all saddles will be exchangeable -- there are a few rare (and very expensive ones) you wouldn't encounter unless you were looking for them which can't be exchanged.
So yes, almost surely if you buy a new saddle you can use it on another bike.
Note that some saddles are marketed as "road" or "mtb" - the mtb ones are possibly more durable, but this is a marketing distinction -- they are not separate standards.
That being said, there are a few things to note in your question:
1) Some discomfort when starting to ride road bikes is normal - the saddles are often harder than what most people are used to, so they need to get used to using their sit bones.
2) Some seatposts don't have much adjustability for saddle angle and position. Thus, a poor seatpost in this respect may make a good saddle for you not feel good. Brooks saddles, for example, are somewhat notorious for needing a decent amount of adjustability in the seatpost.
3) Make sure your bike is fit to you - if the saddle is in the wrong place (or other aspects of the fit are wrong), it won't feel good.
4) You may need a different saddle (as you assume).