ENT reinforces the idea that all Klingon ship types are given avian designations. ENT doesn't clarify who's giving those designations, tho.
It's quite possible that the bird motif is a Vulcan thing, and that the Klingon ships are known as "Warbirds", "Birds of Prey" and "Raptors" in ENT because that's how the Vulcans have classified them and how they have been described for Vulcan's Earthling allies. The native Klingon designations might be far less poetic - perhaps the "battle cruiser" and "attack cruiser" designations used for some of their designs are closer to the native ones?
This Vulcan bird fetish would arguably have survived on those Vulcans who emigrated to become Romulans. But the fact remains that the only Romulan ship design to have an avian designation is the TNG era "Warbird". There is no Romulan Bird of Prey in canon, nor a Romulan Raptor or Buzzard...
It could also be argued that bird motifs are common among starfaring cultures. Starships may look like flying ships to us Earthlings - but others may simply see them as flying things, without the awkward watery connection.
Personally, I prefer to think that Vulcans/Romulans had the bird motive first, and infected it on us humans in the 22nd century - and that ever since those times, the designation "Bird of Prey" has gained a very specific meaning, perhaps that of an atmospheric-capable assault scout (something Starfleet doesn't have), while "Warbird" has become a similar special term, perhaps with the meaning of an Explorer-sized or larger warship (which Starfleet also apparently doesn't have).
That's how current naval designations were born, too: one nation's sometimes nonsensical or sentimental terminology caught on abroad, so that suddenly everybody had "destroyers" for WWI or "frigates" after WWII.
The implication now is that the aliance the Klingons and Romulans had way back during the third season of TOS (The Enterprise Incident) made for the Klingons to have cloaking technology.
Hmh? In ENT, Klingons were the first shown receiving the secret of that tech, in "Unexpected". Romulans were shown possessing the tech quite a bit later, in "Minefield". It may originally have been somewhat likely that there existed an alliance that gave Klingons cloaking tech - but current evidence may actually be against such an alliance ever having existed. The Klingon tech in Romulan hands in "The Enterprise
Incident" may have been war loot for all we know.
As for the Federation not having any, canonical evidence states that the "Treaty of Algeron" forbade the Federation of developing a cloaking device.
...And canon doesn't establish when this treaty was signed. We only know the Feds and the Romulans were signatories, so it has to be after the first encounter with the latter. And it has to be before the early 24th century, because Picard argues the treaty has kept the peace since then. But the signing date could well predate TOS.