As a researcher doing data-stuff: there actually is a somewhat objective way to answer this! I don't know the answer to the question itself though... and the method is quite boring

Usually how data scientists do this is to first collect a bunch of data... let's say we have a 200~300 question comprehensive survey about ppl's political beliefs. This survey would have a dimension of 200-300. We can include all of them but they would offer diminishing information (& is very confusing), so usually people trim it down to the most important dimensions only. We then apply dimensionality reduction/manifold method to reduce highly similar dimensions. I think in social sciences people call this factor analysis. Usually in my field people do PCA followed by UMAP, social scientists I think may do something differently but PCA is quite universal

Then researchers will be able to tell a few mathematically identified dimensions that contribute the most to the results. Say if the first dimension contributes 70% of the variation of people's differences, and the second dimension another 25%... then we would have a 2-dimension model that can explain 95% of the differences and would be good enough. If the first dimension only 10%, second 8%... then a good model will need a lot more dimensions. This doesn't tell what the dimensions are though, that's up to the researchers to identify. If all of these work well, we'd have a simple, N-dimension model suggesting how people's political beliefs are... and some of these might not map to what people would intuitively think of

Unless I'm mistaken, Big Five personality traits is developed this way for example... About politics, I found a 2013 research article that suggested two political dimensions: economic and social ideology

I guess this doesn't quite answer the question... it just states how political dimensions (or any dimensions in data fields, really) came from, and the fact that there's an old paper suggesting a two factor model of economic + social ideology. I don't know how many dimensions are sufficient for politics, not to count for the fact that different countries/cultures treat this differently