I heard someone say this recently.

Problem:
The number of the possessive pronoun does not match the number of the subject.

Explanation:
“Each of us” is singular (the speaker is considering “us” one at a time), so the possessive pronoun in front of “own style” should be singular.

The possessive pronoun “our” is plural and therefore does not match the number of the subject (“Each of us”).

I believe that the speaker of the statement, in ignoring the number of “Each of us” (singular) in favor of the number of “us” (plural), was distracted by the fact that “us” and “our” usually go together (are both first-person plural pronouns) and the proximity of “us” to the possessive pronoun in the statement.

“Each of us has their own style.” would be incorrect, too, because “their” is plural and also is inconsistent with the singularity of “Each of us” (the subject of the sentence).

Given that the statement refers to a group of people, the correct possessive pronouns would be the singular ones “his” and/or “her”. The mental impediment of combining “us” with “his” or “her” in a statement seems to be that “us” is in the first person whereas “his” and “her” are in the third person. One must appreciate that “Each of” in front of “us” forces the listener to consider the third-person individuals in the group.

Solutions:
“Each of us has his own style.” — for an all-male group
“Each of us has her own style.” — for an all-female group
“Each of us has his or her own style.” — for a mixed group