Evening,
All storage mounts use a unique identifier uuid. The datastore name you see if a user-friendly name creates for you (as long as you are on 4.0 or higher). So the answer is you can rename them without any effect. The datastore name is not used by vmware it's used by you and any scripts you create.
So - can they just "rename" these datastores? without risking any problems?
Yes you can rename them without any issues as long as your scripts / documentation / automated processes that are user created don't need them to be named something.
does the datastore name have to be the same on each host?
No because vmware uses the uuid which is the same on each host.
is it possible to rename the datastore on host 1 (so maybe it becomes host1_nfsvol1) and then on vol2 also (so that it is named host2_nfsvol1)?
You can but if they are shared between host1 and host2 your might want the name to be the same to avoid confusion. It's a personal choice.
Please let me know if you have any additional questions.
Thanks