Quite possible, but it refers to the simple political compass; which by the same definition is never accurate (or, binary thinking need not be wrong if you accept that it grossly simplifies).

If it's simplified to the point of being unusable, it's wrong.

There's really no way around that.

At least the Political Compass actually deliberately refers to things which are in deliberate binary opposition. It is literally impossible to be both Authoritarian and Egalitarian. They are definitionally opposed.

It is obviously possible to be both strongly Game and strongly Toy. There is no natural definitional opposition there.

I've seen Factorio, but not PvP. On what is PvP based- real time and you compete who will have more resources and / or destroy the opponent?

The usual PVP mode is a race to see who can get the first rocket fired into orbit. In some sub-modes, all of the teams competing are on the same physical map in different locations and blowing up your enemy's factory facility is a perfectly valid strategy and in others different teams start in the same place on duplicated maps in parallel shards.

The toy aspect of Factorio is then lessened as it now has a goal and measurement, whereas before we could conversely say it had only a direction. With goal and measurement it becomes a game- it is a system of rules the players compete to navigate in the most efficient manner.

Facts not in evidence, Your Honor.

You are making the assertion that toys cannot have both goals and measurements, which flies in the face of actual experience. Some don't, some do. In some cases manipulating the goals and measurements are exactly the mechanism of the toy.

If you mean to say that the introduction of any metric into any environment changes it from a toy to a game, entirely eliminating the toy-nature, then you're going to have to seriously argue that Minecraft turns entirely into a game only when I start building a nerd poll to see how high I can go – and instantly reverts if I jump off, but only so long as I am not curious as to how far I can fall before I die on impact.

This is silly. It's unreasonable to contend.

The toy aspect of Factorio is then lessened [...]

Except, of course, you have defined these things as strictly in opposition and a quality of the entity, not of engagement with that entity.

So which is it, toy or game?

You've effectively defined yourself out of making any kind of reasonable assessment of a thing. If you purport to have a descriptive system, it needs to be able to describe. It needs to be able to describe what it says it describes.

This doesn't.

There are a lot of ludological metrics you could introduce to talk about how someone is playing with a thing at any given time, but trying to do so with a set of binary axes which don't use definitional terms which are oppositional is a fool's game.

This is actually part of why GNS was a terrible system in general, because it tried to describe as a fixed entity systems in a way that would act as a predictive measure for behavioral interactions, when the entirety of the systems that they wanted to describe were the behavioral interactions themselves. They got lost in the map trying to find a way to the next city and just couldn't deal with the idea that the map was not the terrain.

This proposed system doesn't even get that far; it just doesn't start with useful descriptors, at least for the kind of presentation you're suggesting.

You mean this?

Yeah, one of those. Though they tend to look better with only a few metrics if you use an odd number. Eight is pretty close to the minimum number of even-numbered elements you can make look good with that kind of presentation.