I dislike the fact that "other" players can have the same ship name. That sort of ruins the immersion factor for some I would wager, but I'll deal with it.
This would just make people fill up all available ship space on all alts with all the ship names they want to reserve. Non-unique ship names may be a little undesirable, but the alternatives are worse.
I do not have an issue with the current setup with respect to other players being able to name their ships the same as mine (I do have an issue with me not truly being able to make a USS Angmar-B without resorting to correct hull identification numbers but incorrect ship name - though an incorrect ship name is minor).
I personally think that the immersion breaker of very rarely seeing another USS Angmar is MUCH less than having to look at a ship name I dislike every day.
That being said - your issue with why we should/could not do it is normal in nearly every other MMO on the planet so I do not really see that as a minus. There are plenty of names out there.
Heck, I almost always have a character named after myself. In LOTRO I was irritated because all my choices for Brian were "taken" - after an update allowed a server wide search on names I found out that my name 'Brianwald" was the *only* "Brian" prefaced name out there - my craptacular next to last place winning choice of "Brianor" was rejected not because someone else had thought of it, but because the name filter rejected it.
I find the latter MUCH more frustrating than the system we have in STO. Indeed, I find that the idea of letting me have whatever base name I want and then try to make said name unique MUCH more immersive than nearly any system I have otherwise seen. In the real world there is often name overlaps. While not so much in starfleet ship naming, of the things that starfleet doesn't do (say, have all 40 of the people in an instance all have followed the exact same story) this is a small non-cannon immersive idea that geatly works out in other game play areas.