I doubt that many Babylonians or Greeks or any others who cared about such things ever thought Hesperus and Phosphorus were different objects any more than we think the Morning Star and Evening Star are different. Is there any evidence apart from the use of two names?
I think we today underrate what it meant to see the stars clear and bright almost every night and at any part of the night you were awake outdoors -- and with many herdsmen and sailors and revelers and soldiers awake outdoors much of the night, besides people deliberately studying the night sky.
Every slightly interested party knew the circumpolar stars are up all the time, and just get blotted out by sunlight during the day. They had every reason to believe the adjacent stars that rise and set in the North are travelling 'behind' the earth from the time they set until they rise again -- and indeed that the whole sphere of fixed stars exists all the time and moves around.
They knew that Hesperus, when it is seen, moves away from the sun and then back towards it. And they knew that shortly after Hesperus disappears towards the sun, Phosphorus appears on the other side of the sun. Of course stars on one side of the sun are seen just before dawn, moving towards the sun until they disappear and then reappear on the other side just after sunset.
All the major wandering planets obviously move past the sun sometimes, appearing just east of it just after sunset at times, with the sun approaching, disappearing as the sun passes it, and and then appearing just west shortly before sunrise. That is the moon, Mars, Jupiter, and Saturn.
I doubt any great number of people who cared about the question ever thought Hesperus was an exception which ceased to exist each time it moved towards the sun and then Phosphorus appeared, or that Hesperus suddenly went "somewhere else" where we could not see it. These were two names for two roles for one star.
Kuhn's book Copernican Revolution makes these points about the plainly visible motions of the stars -- including the wandering stars. Well, that is plainly visible before artificial lighting, in parts of the world with generally clear skies. He has a nice diagram explaining how people could not possibly have missed the constant existence of circumpolar stars, or the very similar motion of the stars near them that do rise and set.
B. L. van der Waerden studied Babylonian astronomy and simply reports they knew Venus as a planet -- long before the Greek astronomers that we know. (You can see this in his "Babylonian Astronomy. II. The Thirty-Six Stars," Journal of Near Eastern Studies, Vol. 8, No. 1 (Jan., 1949), pp. 6-26) available at www.jstor.org/stable/542436 .
A lot of sources on-line say the Egyptians and Greeks thought the morning star and evening stars were different stars. But what they actually show is that those people associated different gods to the morning and evening stars. The Egyptians also associated Khepera to the rising sun and Atum to the setting sun, without thinking the rising and setting suns were two different things.
I do not see any evidence of a time when people tried to individuate "stars" as objects but thought the morning star and evening star were two different ones.
Added: Plato's Laws 821-822 has the interlocutors discuss the "impious error" that Greeks make when they believe "Hesperus and Phosphorus and the other planets" wander rather than following a single course. One character asserts "The truth is precisely the opposite: each of them always travels in a circle one and the same path." The other does not disagree. The Greek is no more explicit than the English given here. It does say anyone thinks "Hesperus and Phoshorus" are two things, and anyway it is discussing an error without saying which Greeks make this mistake. To believe Hesperus follows one circular orbit you would have to believe half of that orbit goes under the name of Phosphorus.