unless you are cool with DMs only handing out unique magic swords to whoever they please instead of them being something each player had the ability to go and get for their self. I would not particularly prefer that over having the hard-to-explain copies that perhaps some other character mine knows having one of the same.
As a matter of fact? Yes, yes, I am. And I say this as a player who is less than likely to get a unique weapon because it's pretty clear that certain DMs don't particularly like me, and the rest just don't know or have an interest in me. I say this as a player who's been ousted from a DM event that the DM initiated TO me, because the DM wanted to give it to someone they liked better. On this server. I say this as a player who is the least likely to benefit from this, simply due to politics, and why do I say this? Because it makes sense - even as a PW server, if not especially as a PW server. Persistent World doesn't mean 'concede to everyone' either. And on Ravenloft being the guy with the unique weapon means you earned your merits, rather than 'oh hey, after farming this place fifty times I stumbled on it'.
Also, this isn't a case of 'whoops, there's more than one magic sword'. This is a case of 'hey, you know that one of a kind, ultra-legendary weapon that spans history itself and everyone knows about? I gots three!' Even then, if you wanted magic swords of this type, there's a simple solution to that: make them a particular brand of magic sword. You know what a magic sword can be? Something as simple as a +1 enhancement to it, and you know what that sword is called? 'An enchanted sword'. It has no legacy attached to it, it's just a particularly magical blade. You know what a magic sword that does 1d4 fire damage is because it has an enchantment placed on the ruby in its pommel? 'An enchanted fire-ruby sword'. No legacy, but its functionality is there.
If your point is that 'oh hey, who cares, it's just a name', let's put that to the test and I'll go ahead and put in an item request for scimitars with the same names as Drizzt's swords, Twinkle and Icingdeath. If it's 'just names', then it shouldn't be a problem in the slightest, right?