Because they traded technology
From the Memory-Alpha write-up for TOS: The Enterprise Incident
Kirk then inexplicably orders helmsman Sulu to take the Enterprise
across the Romulan Neutral Zone and into Romulan space, where it is
quickly surrounded by the Romulans – who are now using Klingon-style
An effective cloaking device, meanwhile, was passed from the Romulans to the Klingons at about the same time. The last mentioned episode took place in 2268, while a working Klingon cloaking device was present in the episode TAS:The Time Trap (that is the animated series), which took place in 2269.
A writer for Star Trek III: The Search for Spock explicitly stated that he used the previously Romulan Bird-of-Prey designation for the a Klingon ship on purpose. According to writer Harve Bennett:
"I didn't change their ship, because I remembered a piece of trivia
that stated there was a mutual assistance military pact between the
Klingons and the Romulans for an exchange of a military equipment."
(Starlog #103, February 1986, p. 17)
This short-lived Romulan-Klingon alliance was never explicitly stated in the original series, but was evidently supposed to feature more prominently in the never-made Season 4; this is where the writers picked up the ideas that eventually lead to Klingon cloaking devices in the animated series and the movies.
In any case, the swapping of ship designs and cloaking technology lead to some convergence in ship design, particularly around the bird motif.