By core rules, likely in none of them (maybe in 1e and 4e)
Below is the perspective on core rules from each edition. It may however be possible that there are further prestige classes and splat books for the various editions that enable it. (Emphasis in quotes mine).
Very likely not in AD&D 1e
Shapechange states (PHB p. 21):
Ability to change form up to three times per day, actually
becoming, in all respects save the mind, a reptile, bird or
mammal. (...) The size of creature form assumed can vary from as
small as a bullfrog, bluejay or bat to as large as a large
snake, an eagle, or o black bear (about double the
weight of the druid).
1e had no creature types, but the owlbear (MM, p. 77) description states:
If encountered in their lair there is a 25% chance that there will be 1-6 eggs (20%) or young (80%) in addition to the adults (...) Owlbears have brownish-black to yellow brown fur and feathers. The 1,300 to 1,500 pound males will be the darker colored.
It is not clear if the owlbear counts as a mammal, given there are zootic weirdnesses like the Platypus. 2e, which is very close rules-wise calls it out as a mammal, which would suggest yes.
The other issue is the weight. An adult brown bear, weighs about 500 pounds, but females can be a lithe as 335 pounds, or about 2/3rds of that, so a female owlbear could weigh as little as 850 pounds. By the height and weight table in the DMG (p. 102), the maximum weight for the heaviest type of player character, a male human, would be 175 + 60 pounds, or 235 pounds, and twice that is 470 pounds, far lower. So, unfortuneately there is no way to transform to an adult owlbear. Maybe an owlbear cub.
I strongly suspect that going by Gygaxian sensibilities, it was not intended to be possible.
Not in AD&D 2e
PHB, page 37, Druid class description:
He gains the ability to shapechange into a reptile, bird, or mammal up to three times per day after he reaches 7th level (...) The druid can only assume of a normal (real world) animal in its normal proportions
The owlbear in the Monstrous Manual states:
They are warm-blooded mammals, but lay eggs.
but they certainly are not a real world animal.
Not in 3.0 core rules, but possible with prestige class
The SRD states
At 5th level, a druid gains the spell-like ability to polymorph self into a Small or Medium-size animal (but not a dire animal) and back again once per day.
The owlbear in 3.0 is a large beast (MM, page 148, at least the printing I have access to). This is differnt a type than animal in 3e, which is its own type (MM, p. 13). As Molot points out, you would be able to do it with a splat book prestige class.
This later upgrades to Large animals and Elementals.
Not in 3.5 (but see options from splat books)
KRyans answer as well as Molot's provide options that allow to do it, for example with the epic feat "Magical Beast Wild Shape" from Complete Divine, p. 90. However, I am focused only on the core rules.
The SRD states:
At 5th level, a druid gains the ability to turn herself into any Small or Medium animal and back again once per day. Her options for new forms include all creatures with the animal type.
At higher levels this upgrades to larger and smaller animals, plants, and elementals, no other types.
The owlbear in 3.5 is a magical beast.
Maybe in 4e, for a medium-sized owlbear variant
Page 83 and 84, PHB II:
The wild shape power lets you assume a form of your size that resembles a natural or a fey beast, usually a four-legged mammalian predator such as a bear, a boar, a panther, a wolf, or a wolverine.
Your beast form might also be an indistinct shape (...) You might also resemble a more exotic beast when you’re in beast form: a reptile such as a rage drake or a crocodile, or a fantastic beast such as an owlbear or a bulette.
(Thank you to @KRyan for pointing this section out).
Wild shape allows a character to change between humanoid form and beast form. (...) You can choose a specific form when you use wild shape to change into beast form. The beast form is your size, resembles a natural beast or a fey beast, and normally doesn't change your game statistics or movement modes.
The owlbear in 4e is a large fey beast, so the change would not work for a medium or small sized creature like all PC races in the PHB I and II are. (Thanks go to @Draconis for providing the expertise here -- I've never played 4e myself).
However, owlbear is given as an explicit example for a form to resemble, even though based on size you cannot change into an actual owlbear normally. Maybe if the druid was enlarged magically first? Or maybe you could take the form of an owlbear, but it would be smaller, like an owlbear cub (compare to 1st edition). Medium-sized owlbears are mentioned a few times in 4e material, so it seems quite possible. (Thank you again to @KRyan).
Not in 5e
As you state in the question.