No, there are no weapon size rules in 5e beyond Heavy weapons being off limits to size Small races.
There is no feature like that in the Elemental Evil Players Companion entry for the Goliath race. Powerful Build only applies to carrying capacity and the weight limits for moving things around (push, drag, or lift).
There are no rules for weapon size in general in 5e, thus there are no size large/size giant weapon rules with wielding penalties and damage bonuses as in previous editions. PCs wield weapons appropriate for their size with no restrictions except Small creatures being unable to wield heavy weapons.
Size
Characters of most races are Medium, a size category including creatures that are roughly 4 to 8 feet tall. Members of a few races are Small (between 2 and 4 feet tall), which means that certain rules of the game affect them differently. The most important of these rules is that Small characters have trouble wielding heavy weapons, as explained in chapter 6. - (PHB p. 17)
Heavy
Small creatures have disadvantage on attack rolls with heavy weapons. A heavy weapon’s size and bulk make it too large for a Small creature to use effectively. - (PHB p. 147)
Goliaths already have some fairly strong racial features and do not need a permanent racial damage buff.
As RAW, no, it does not
Being considered a size higher for carrying capacity and push, drag, or lift force is not the same as being a size higher for all purposes, including grappling.
You count as one size larger when determining your carrying capacity
and the weight you can push, drag, or lift.
As RAI, maybe
The loop hole is that the PHB has no reference on why and how the grapple-er can move without speed penalty if it is two size higher, and the DMG does not touch the subject. There are two potential scenarios for why a grapple-er one size or lower has it speed reduced, though; the grappled creature have enough leverage to make the movement difficult or that the grapple-er cannot carry comfortable enough the grappled creature as in "difficult terrain", and that the weight of the creature is the deterrent.
In the first case, since Powerful Build does not increase per-se the size, it is safe to assume that it does not help to increase the comfort or reduce the leverage and, such, the speed is halved.
In the second case, where is the weight that maters, things change. Powerful Build improves the carrying, dragging, and lift capacity, therefore the grapple-er creature does not have its speed halved.
Which one it is
The only evidence I found is in favor of is that weight is the reason for the reduction in speed. The first part is in the text of moving a grappled creature (emphasis mine).
When you move, you can drag or carry the grappled creature with you,
but your speed is halved, unless the creature is two or more sizes
smaller than you.
As far as I can tell, there are no rules on how you drag or, particularly, carry a grappled creature. You can carry over you head a grapple creature as if it were a sack of potatoes. In PHB 176 there is a segment that shows how can you drag, carry and lift something in particular and, as shown, it is a STR and weight contest in which Powerful Build should work as intended.
Push, Drag, or Lift. You can push, drag, or lift a weight in pounds up
to twice your carrying capacity (or 30 times your Strength score).
Bottom line
At the end is the DM that has the last word. Talk to your DM and present the evidence. If you are the DM decide what it is best for your campaign. If a particular decision is more powerful that you expected it is in your right to take back the decision.
Best Answer
Carried? Yes, as cargo. As a mounted rider? No.
A reasonably strong Centaur should have no issue physically carrying a Medium creature.1 With only a Strength score of 10, a centaur has a carrying capacity of 300 pounds, increasing by 30 pounds for each point added to their Strength score. But being carried in this way does not confer the benefits of having mounted the centaur, which seems to be what is being asked, as evidenced by use of the word "rider".
The centaur's Equine Build trait specifies which things the centaur is considered Large for:
The rules for mounted combat say:
Since being mounted is not mentioned in Equine Build, you are still considered Medium for the purposes of what size creature can mount you - Small or smaller.
This is definitely rule-as-intended. In the Unearthed Arcana version of the Centaur race (.pdf link), the Equine Build trait included the following text:
This portion of the feature was removed for the official release of the Centaur playable race in Guildmasters' Guide to Ravnica.
What can I do while being carried-as-cargo by a centaur?
There is an optional rule in the Dungeon Master's Guide that can give us some guidance, but I must emphasize, everything that follows is entirely at the DM's discretion.
The DMG has an optional rule for climbing onto a bigger creature (p. 271):
So it will be up to the DM to determine if a centaur is "suitably large". It also has some details about contested ability checks, but those shouldn't be an issue for a willing centaur.
From here, it is up to the DM. If your table is going to go this route, this should be viewed as a house rule guided by an optional rule from the DMG, and your table should decide together what it looks like, and write it down so it can be consistently adjudicated.
Personally, I would recommend against this, as this is clearly contrary to the intent of the rule.
1 Thanks to user SirTechSpec for pointing out this important distinction.