What is a unit of wealth? Took us a while but we figured out fiat currency.
That being said, I’d argue “2001 dollars” is a unit, not “dollars”. You’ll often see data in econ journals cite something in that type of unit.
Another angle on this is that comparing wealth across time is nontrivial (and the money isn’t to blame). How do you compare the value of a iPhone in 2020 to a landline in 1980? It’s possible, but nontrivial. Our units (in some messy imperfect way) reflect that (and other factors).
Why though? Why dollars in a particular year, rather than dollars in general? Sure the value of a dollar changes from year to year, but the value of a dollar also changes constantly in any given year too. Over the past year, the value of a US dollar relative to a Euro has fluctuated by more than 10 cents. In fact the relative values of currencies fluctuate constantly, near-enough to instantaneously. But this isn't reflected in the pricing of most consumer goods; the price of common household goods in a grocery store might vary a few times a week, but not thousands of times an hour. So in one context the value of a dollar seems to vary constantly, but in another context the value of a dollar is substantially more stable. So a dollar doesn't even have a set particular value at any instant in time; it depends on what kind of store you're standing in.
If happiness is a metric with units even worse than this, then I see little hope for any scientific study of happiness.
Say what you want about stellar kinematics, but measuring meters simply is not so fraught as this.
(If you’re asking about the units for wealth: USD, Euros, etc)
So it's not like it can assess this number for an individual, at least not without first comparing and contrasting it with every other person in some sample group.
In other words, a true unit of happiness, would allow a completely independent measure of any one individual's current state of happiness. Imagine something like a thermometer for happiness, you point it at someone and it reads out the units of happiness (over-simplifying the application to make the point). Therefore we're not even close to knowing what this is or how to measure it, and the way we really communicate around it is by using our own subjectivity of it and making correlations.
The ladder question (a common tool for measuring SWB) is this:
Please imagine a ladder with steps numbered from zero at the bottom to 10 at the top.
The top of the ladder represents the best possible life for you and the bottom of the ladder represents the worst possible life for you.
On which step of the ladder would you say you personally feel you stand at this time? (ladder-present)