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@LSpice - the scan of that very published version is available on OEIS, as indicated in my question. Somehow, I am not sure that linking the paywalled version serves any purpose...
The question still has no clear structure. You are not new on MathOverflow, you surely mist have some ideas of what it means to ask a structured and readable question.
Terminologically, a commutative nonassociative structure satisfying your "property A" is often called "a commutative loop with the inverse property" - probably doing some search for this term in the existing literature you will be able to find some information.
what I would like is for the question to have a bit more structure - definitions, references, motivation - the way it is stated currently, it is all over the place.
(continued) In the analytic context, you talk about convergence in $\mathbb{R}$ for specific values of $x$. In the context of formal series, you look at convergence with respect to the topology given by powers of the maximal ideal, which forces the coefficients of a convergent sequence of power series to stabilize, so sums have to be finite. Until you have (much more) clarity about what you want, your question is too vague and too broad.
@MaxMuller when you work with power series you have to make a clear decision of (1) what is the full set of power series you are working with, (2) what are the structure operations you are considering on this set, and, if you wish to use infinite sums or products as structure operations, (3) what kind of convergence you work with. (continued in the next comment)
@MaxMuller the difference between power series (an analytic object) and formal power series (an algebraic object) is that in the formal context you are only permitted to perform algebraic operations - in particular, computing infinite sums of real numbers is not an option. So in the formal context, your formula (1) makes sense, and your formula (2) does not.
@IgorMakhlin - interesting! I was not aware of these papers. Particularly arxiv.org/abs/2111.08754 might be a good candidate for a conceptual answer to the OP's question.
@IgorMakhlin the ring is graded, and the $sl_n$ action is going to mess it up, so something non-trivial will have to happen. I don't know what it could be.
@SamHopkins here is a speculative idea. Coordinates of the Plücker embedding should separate different points. If we wish that the embedding is defined in a universal way, we perhaps should wish that it separates points over $\mathbb{F}_1$, which naively makes one think that the number of coordinates should be given by the number of $\mathbb{F}_1$-points, that is $\binom{n}{d}$. Someone who is well versed in geometry over $\mathbb{F}_1$ can say if this can be made rigorous, but as a heuristics I certainly find it rather useful.
