Yes.
The soulknife ability clearly states that if a soulknife has Powerful build the knife it manifests is larger to match the ability. So you would deal increased damage due to having a larger knife than say.. a regular human.
By design this also means if you can find a way to get Enlarge Person to increase your size, then you would also benefit from the soulknife damage increase granted by powerful build as well.
Form Soul blade states:
A soulknife must choose the form of her mind blade at 1st level. She can either form it into a light weapon, a one-handed weapon, or a two-handed weapon. Once chosen, her mind blade stays in this form every time the soulknife forms her mind blade. The light weapon deals 1d6 points of damage, the one-handed weapon deals 1d8 points of damage, and the two-handed weapon deals 2d6 points of damage. All damages are based on a Medium-sized creature wielding Medium-sized weapons.
This constrains the Soul blade to a certain type of weapon, such as longsword, shortsword, trident, etc. Whichever you choose when you first manifest it. The size changes based on your own size, which may have variable scaling.
Technically speaking, yes. The looted weapon will still deal extra damage.
Big monsters typically wield oversized weapons that deal extra dice of damage on a hit.
The "that" is pretty clear - the weapons deal extra damage, not the monster.
However, you should be extremely cautious in making these rules (and weapons) accessible to players. The rules on creature size are in the PHB, but these rules are in the DMG. They're not meant to be easily accessible to players.
A player who is making a conscious effort to gain advantage on their attacks can do so fairly easily. That makes wielding oversized weapons a pretty nice option for any player who wants to build around it. At the very least, if you're planning to allow this, consider implementing the suggested rule that it is impossible to use a weapon two sizes too big.
Best Answer
DM decision, but probably 2d8+1d4+STR damage at disadvantage
Lager weapon means more damage
The Ogre's Greatclub attack does 2d8 + 4 damage according to its statblock.
The rules in the DMG say that:
So the greatclub the orge is holding is extra big thus accounting for the damage.
A normal greatclub does 1d8 damage, but this oversized greatclub does 2d8 damage.
Enlarging the PC and weapon
Enlarge says:
When a medium-sized PC carrying a large-sized weapon is enlarged one of two things will happen. The DM will have to decide which one is the interpretation that they prefer since there is some ambiguity here.
Large PC + Large greatclub
Since enlarge says the "weapons also grow to match its new size" one could interpret this as meaning that since the greatclub was already large that it does not grow at all (since large "matches" the PC's new size). Since the greatclub did not technically grow in size, it does not get the bonus damage from enlarge either. On the other hand, since the PC and weapon size now match, there is no disadvantage.
An enlarged PC using a large greatclub would do 2d8+STR damage (no disadvantage)
Large PC + Huge greatclub
One could also argue that this spell was not written with mismatched character/weapon sizes in mind and that the intent of the spell is probably that everything just increases in size proportionately. In this case the character and weapon both increase one size category. Since their size stays mismatched the character will still have disadvantage wielding the huge greatclub.
An enlarged PC using a huge greatclub might do 2d8+1d4+STR damage (with disadvantage)
Option #1 vs #2: bigger isn't always better
Given that the difference between the two options is a relatively small 1d4 damage, I personally would rule for whichever the players found more fun and apply that consistently.
Both options are pretty close in terms of damage output: option #2 would average to 13+STR (5+5+3) but will have disadvantage and option #1 would average out to 10+STR (5+5) and will not have disadvantage. #1 has the benefits of being more in line with the literal wording of the spell effect as well as not having disadvantage which can reduce damage output. Either way you choose, it shouldn't make a huge difference in terms of power.
Warning for DM
It is worth noting that monster-creation math and PC-creation math is intentionally different/incompatible with each other. As a DM be careful when you give a PC a monster weapon that you are giving them only the weapon part of the damage without any of the bonuses that comes from the monster itself. It is very clear in the ogre example, but may not be so clear with other examples.