Critics argue that there are limitations to the scope of Big Five as an explanatory or predictive theory. It is argued that the Big Five does not explain all of human personality.

If people claim that it is an all encompassing model, then that would be a serious criticism. I don't actually know if researchers claim that, but it seems unlikely to me.

The methodology used to identify the dimensional structure of personality traits, factor analysis, is often challenged for not having a universally-recognized basis for choosing among solutions with different numbers of factors.

This is a weaker criticism than it seems. It stems from using classical stats. Latent factor (such as FA) models are less confusing from a Bayesian point of view. The particular indeterminacy they are talking about disappears when you specify a model and prior more clearly and try to update in a principled manner. A better criticism would be that the results of personality models might be sensitive to model assumptions. However, that sensitivity is an empirical matter and isn't established by referencing the indeterminacy of FA. Another good criticism of Big Five research might be that factor analysis is not a good tool because it could be hard to figure out how sensitive your results are to the particular analysis.

Another frequent criticism is that the Big Five is not theory-driven. It is merely a data-driven investigation of certain descriptors that tend to cluster together under factor analysis.

This doesn't seems like a criticism to me.