No player including the catcher is allowed to block the base unless he has the ball. It would be defensive interference and the runner would be safe. The rule is generally not enforced if the ball and the runner arrive at the same time.
The reason home plate is somewhat different than second or third is because a runner does not need to remain on home plate where at second and third, overrunning the bag could allow the runner to be tagged out.
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A catcher can only block home plate if they have possession of the ball. This is true for any base. However, if the defensive player does not have possession of the ball and they block the plate, the runner is granted the base due to obstruction.