I would call a half-ogre a monstrous race choice.
In fact, I did: "abilities due to their monstrous nature."
You missed the point. I was not trying to say that monstrous races don't need to try and be balanced. I was noting that the monstrous races did things like step out of the normal assumptions of class design in order to play with decisions that are either a) contentious or b) do things that are not commonly done, such as use mechanical penalties for flavour. You have chosen to emulate their design. Cool. The penalties to Int, however, do not balance the half-ogres benefits. They are merely flavour (if you want to take semantic issue with the word fluff).
Intelligence ability scores are mechanics not fluff; they may be less significant than other scores but mechanics nonetheless.
I don't see the "internal logic" of a large half-ogre/half-bugbear getting a reduction to Strength and Constitution when both of its parents had increases in those stats or an increase in Intelligence when a half-ogre only has a 7.
If you want to argue semantics, we can use the word "flavour" instead of fluff. In terms of a larger philosophical debate, I don't believe fluff and mechanics are as divorced as you think they are. I am not, however, interested in having that debate here. If you are interested in that subject, you can read such texts as Wardrip-Fruin's Expressive Processing. The point is, the penalty might be a "mechanic" insofar as you are defining it, and a penalty insofar as it reduces some mechanic, but it is a meaningless reduction that won't be felt by the builds that the rest of your mechanics support. As a result, it doesn't do anything to balance the race. Your penalty to "Intelligence" should be thought of as nothing more than flavour. It does nothing to balance a race which, when combined with its optimal class choices, will have a +2 to its primary and secondary stat choices and a +1d4/per attack to its primary source of damage. You are right, though. In the case of a Half-Ogre, a penalty to Strength or Constitution would not be internally consistent. I was not suggesting that you do that. I was merely noting that only a penalty to Strength or Constitution could act as a balancing factor for this race.
I think the problem with players asking why their half-ogres are different than the NPC race applies here as well.
Perhaps a Constitution reduction is needed as the MM half-ogre only has a 14, giving on one +2 score bonus; that's an easy fix.
Yea. So long as the race has no more than a +2 to one stat and a +1 to, at most, 2 other stats (and that is pushing it considering its other powers; I would really suggest a lone +2 stat adjustment if its going to have a constant +1d4 damage bump), you might be able to make it work.
No worries. As a monstrous PC, you wouldn't have to.
You are, again, choosing to miss the point. I would, for example, allow all of the new PC race choices from Volo's guide. I would not allow your race at the table in its current form. You asked for constructive critique. I am trying to give it to you. You have something workable. It still, however, needs more tweaking. I am trying to provide you with feedback as to what is currently not working.
That seems to defy "internal logic" again: is there an actual size difference between "large" and "oversized"?
We're talking about an 8' tall 450 pound half-ogre with a large long sword compared to an 8' tall 450 pound bugbear with a regular long sword; would there be a longsword in between those two weapons?
Honestly, considering the fact that you are working with the size stats of a bugbear, I would probably try and make the creature medium sized. I would probably then give it a racial feature that looked something like this: "Oversized: for all intents and purposes, the half-ogre counts as a large sized creature. When it wields large sized weapons, however, it deals the same damage as the weapon's medium sized equivalent +1d4. The half-ogre may still wield normal weapons without penalty." You, however, seem to really want to keep the "large" size as is. Given that fact, I have given you a few constructive suggestions. If you take the "oversized weapons" suggestions, then yes, you would be saying that there is a 1/2 size increase between medium and large weapons called "oversized," and that is what the half-ogre can use. If you take the second option, you would simply be letting the half-ogre use large weapons, but would be stating that it can't do as much damage with them as normal large creatures. What you don't, however, want to say is that large weapons (in general) do +1d4 damage when the rest of the game says that large weapons deal double the normal damage dice. It creates rules inconsistencies; it is not internally consistent.
I'm assuming you meant half-orgres instead of half-giants.
I think something like that would work, but the MM large-size half-ogre does get the double damage die and is only 8' tall; the MM large-size ogre is 9' to 10' tall and also gets the double damage die.
Yep. By choosing to create a half-giant PC race that doesn't get double damage dice, you have already created a minimal level of internal inconsistency. I don't like it. I wouldn't want to do that to my games. You, however, seem set on that choice, so I am trying to provide you with constructive criticism designed to minimize the issues wherever possible. As I said earlier, the only way I see to faithfully create a race like the half-ogre for PC would be with a few levels of a front end "race-as-class" design. After experimenting, it is the only thing that worked well for me. You are not me, though, so I am trying to give you constructive criticism within your desired constraints.
TL;DR--your original race's presentation is ok, but it is still a little too powerful, and it creates one too many rules inconsistencies in one area. This race should not have a +2 to both a primary and secondary stat. Overall, it is strong enough that it should not have more than a +2 to a primary stat and a +1 to a tertiary stat (not Con). The +1d4 damage bonus is strong enough that you should probably stick to a flat +2 bonus to a primary stat or a +1 bonus to a primary and a secondary stat. You should also avoid defining large weapons as weapons that deal +1d4 damage beyond their normal counterparts, as the rest of the game already defines them in another way.