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One of main characteristics of Five Factor Model is that factors are not orthogonal as in some other personality models.
There are no personality models in which the factors are orthogonal in the strict sense of being at 90 degrees from one another, unless an orthogonal rotation method is used to extract them. I think you meant to say that the Big Five factors are less independent (or more strongly inter-correlated) relative to the factors in other models.
Whether or not one wants to obtain orthogonal factors depends on what one is trying to achieve. In most cases, you want an orthogonal solution if you want simple structure, i.e. you want to figure out which factor indicators unambiguously cluster together to form a relatively independent factor. But in general, factors are allowed to correlate because that just happens to be the "reality" and there is no theory suggesting that they should be completely independent. There are of course plenty of reasons why relatively independent factors might correlate and it doesn't mean that biological bases cannot be found; it could mean that there are biological bases that are shared among traits and there are those that are unique to each trait. For example, the relationship between Openness and Extraversion could in part be due to shared genes related to exploration (I forget the source for this).
This previous answer talks about evidence for a global or possibly two factor model partially explaining the correlations between the Big 5. In general this global factor may reflect some form of positive, well-being, social adjustment type variance.
More generally, having orthogonal factors is desirable from a parsimony perspective. It's useful when predicting other outcome variables. However, orthogonality is not the only criteria. in fact many psychological scales are explicitly hierarchical with global scales accompanied by correlated subscales (e.g., many cognitive measures; the facets in the NEO-PI-R, etc). As @AlexR notes factor analysis can force an orthogonal rotation, but this doesn't mean that scales formed by summing corresponding items yield uncorrelated scales.
I have heard it said that the Big 5 is too complex to have a natural mapping to biological processes. That said, height and weight are correlated and they certainly have a biological basis. So orthogonality is not a requirement. And twin studies support the observation that personality has a fairly large heritable component.
I would like to answer this question from a mathematical point of view.
The word "openness" means something that is real and correlated with some other aspects. Let's assume a 3 dimensional space.
If we would like to choose the other dimensions as perpendicular to this first one, we could simply define them by their correlation with the first one being 0.
The definition is mathematic. Given a vector finding ortogonal ones means finding something that could exist or not in the nature.
Given 5 dimensions which are correlated and really exists (i.e. the big five traits), we could easily find a new space made by one real dimension and 4 dimension ortogonal to the first one which doesn't exist in the reality.
Then we could project every person analisys from the Big Five Space to the second perfectly orthogonal space.
Every dimension of the second space could be also defined as a mix of the dimensions of the first space.
You could ask for a better answer here but I hope something of what I said was clear.
https://math.stackexchange.com/questions/382195/translation-an-orthogonal-transformation
It is often assumed that the five factors are theoretically orthogonal (https://doi.org/10.1093/oxfordhb/9780199352487.013.20). In practice, the the five factors are highly correlated; however, this is usually ignored in favour of the assumed theoretical orthogonality. An alternative explanation is that the correlation has been ignored due to the absence of a better model in which the dimensions are biologically based and strictly orthogonal.
A more recent development posits a two-dimensional taxonomy in which the dimensions (affiliation and dominance) are strictly orthogonal. The biological basis for these two dimensions are identified in the research (affiliation/oxytocin/Putamen) and (dominance/testosterone/Prefrontal cortex). https://doi.org/10.1371/journal.pone.0227877
