As I said elsewhere, this is not a problem of units or types. If it were, then the computation wouldn't make sense.

It is a problem of implicit refernces. The two 1/3 fractions refer to different objects ('wholes') than the 2/6 fraction (and from each other).

The correct equation would have been 1/3 * 3 + 1/3 * 3 = 2/6 * 6. Note that 3, 3 and 6 have the same unit. If they didn't, then this would be meaningless. 1/3 of a meter + 1/3 of a Pascal does not equal 2/6 of anything (or maybe it does equal 2/6 of (2 meters + 2 Pascals) ...).

> The two 1/3 fractions refer to different objects ('wholes') than the 2/6 fraction (and from each other).

That is, precisely, the problem of units. Each object is its own unit here.

> The correct equation would have been 1/3 3 + 1/3 * 3 = 2/6 * 6.*

That's not how you add fractions. The correct equation would have been, 1/3 * 3 + 1/3 * 3 = 2/3 * 3 (or, 1+1=2), if these 3 all truly had the same units. But they don't, so you can't add like that.

> 1/3 of a meter + 1/3 of a Pascal does not equal 2/6 of anything (or maybe it does equal 2/6 of (2 meters + 2 Pascals)

That's the point (but it's 1/3, not 2/6). Also, 1/3 of a meter, + 1/3 foot = 1/3 (1 meter + 1 foot). Different units, but same dimension, so if you know the conversion factor (here, 1 meter = 3.3 feet), you can change it into (1/3 meter * 3.3 feet/meter) + 1/3 foot = 1.1 foot + 1/3 foot = 33/30 feet + 10/30 feet = 43/30 feet = 1.43(3) feet.

You can do the same math with students at tables.

You are trying to look at a different problem. It was absolutely correct that 1/3 of the students at one table of 3 plus another 1/3 of the students at another table of 3 is the same number as 2/6 of the 6 students sitting at the two tables. This is not disputable.

The way you can write this observation mathematically is as I did: ((1/3) × 3) + ((1/3) × 3) = ((2/6) × 6), or 1 + 1 = 2, after computing the fractions. The student's observation was perfectly correct, but he was missing the proper explanation, as it is not about the addition of fractions (it is almost a coincidence that the fractions used on one side of the equation happen to have the sum of their numerator and the sum of their denominators equal to the numerator and denominator of the fraction on the other side - this only happens because we are multiplying the fractions by their denominators).

Sure, you can express this in terms of units and dimensions of you really choose to. You can also express it in terms of different definitions of +, or even of =. It is pretty unnatural to me to invent an ad-hoc measurement unit N1, "number of people at 1 table" and a different measurement unit, N2, "number of people at 2 tables", with the relation 1N2 = 2N1, and then correct the student's formula to 1/3N1 + 1/3N1 = 2/6N2. It is correct, but it is extremely artificial to me.

By far the most natural way to explain it is using the correct mathematical interpretation of the phrase "one third of the 3 people" - (1/3) × 3.

Inventing measurement units to describe exact quantities reminds me of a silly joke from Portal: "computer: 2 + 2 = 10 <pause to wonder if the computer is broken> ... in base 4". You can always find a way to make the formula direct by adding assumptions.