For cross save, the accounts in question are going to need to be linked to your bungie.net profile so regardless of what the "name" is it only the account that is linked to your bnet will have your save.
Your characters are bound to your bungie account. You can have the same name as someone else on battlenet and still have separate characters. You will be setting one platform's account as the primary one and then linking your other systems to that account I believe. So on each system you will have a different name, because it is using that system's API for multiplayer.