I agree that drive letters are a very poor way of organizing mounted partitions, but Microsoft probably can't change it at this point because switching from drive letters to folder mount points wholesale would break assumptions made by many programs - sacrificing legacy compatibility - which they won't do.
When a flash drive is inserted into a computer running OS X, Finder will automatically mount its partitions to
/Volumes/Some Partition Name. If there is already a partition mounted at
/Volumes/Some Partition Name, Finder will mount the partition to
/Volumes/Some Partition Name 1 instead. Nautilus (the default file manager in Ubuntu) does something very similar, except partitions are mounted to
/media/Some Partition Name and conflicts are resolved by appending
(number) to the end of the mount point. Therefore, so long as the label of the partition on your flash drive is sufficiently unique, you can reasonably expect that it will be mounted in the same place no matter which USB port you plug it into or which Mac you use.
As others have already mentioned, Windows also allows you to
mount partitions to folders - which might solve your problem - but I'm not sure if those mount points are persistent. Based on the way that Windows remembers drive letters (which billyea nailed in the first reply), I would guess that it behaves similarly for folder mount points.