This is a common misunderstanding and unfortunately strengthened by the example of 'personality embeddings'. It is easy to understand intuitively why this is normally not the case. If you rotate a vector/embedding space, all the cosine similarities between words are preserved. Suppose that component 20 encoded 'royalty', there is an infinite number of rotations of the vector space where 'royalty' is distributed among many dimensions. Consider e.g. the personality vector openness-extroversion as values between 0-1. Now we have two persons:
[0, 1] [1, 0]
These vectors are orthogonal, so the cosine similarity is 0. Now let's rotate the vector space by 45 degrees:
[-0.707, 0.707] [0.707, 0.707]
The cosine similarity between the two vectors is still 0. Clearly, the direct mapping of personality traits to vector components is lost (as can be seen in the second vector).
Obviously, this is something that you generally do not want to do for personality vectors. However, there is nothing in the word2vec objectives that would prefer a vector space with meaningful dimensions. E.g. take the skip-gram model, which maximizes the log-likelihood of the probability of a context word f, cooccuring with a context word c. Shortened: p(1|w_f,w_c) = 𝜎(w_f·w_c). So, the objective in vanilla word2vec prefers vector spaces that maximize the inner product of words that co-occur and minimize the inner product of words that do not co-occur. Consequently, if we have an optimal parametrization of W and C (the word and context matrices), any rotation of the vector space is also an optimal solution. Which rotation you actually get is dependent on accidental factors, such as the initial (typically randomized) parameters.
Of course, it is possible to rotate the vector space such that dimensions become meaningful (see e.g. [1]), but with word2vec's default objective meaningful dimensions are purely accidental, and the meaning of vectors is defined in their relation to other vectors / their neighborhood.