The interpretation now universal, that the dreamer is running from something they are avoiding in waking life, is psychoanalytic in origin and twentieth-century in date, and it should be attributed rather than assumed.
The general move, that a dream dramatises a conflict the dreamer will not face directly, is Freud's framework, though the chase is not one of his signature examples. The specific and much stronger claim, that the pursuer is a rejected part of the dreamer's own self and that the dream asks for confrontation rather than flight, is Jung's, and it follows directly from his account of the shadow: the qualities a person refuses to acknowledge are, on his model, personified in dreams as a threatening figure, and the therapeutic task is recognition and integration.
That is a coherent theory. It is also a theory, held by a named person, resting on a particular model of the psyche that not everyone accepts, and it should be presented that way rather than as a fact about what chase dreams are.
Competing explanations exist and do not involve meaning at all. The threat simulation hypothesis, associated with Antti Revonsuo, proposes that dreaming evolved to rehearse responses to danger, which would predict exactly this dream's prevalence without any symbolic content whatsoever. Neurobiological accounts of nightmares as dysregulated fear processing make similar predictions.
The honest position is that the dream is common, that its prevalence has several possible explanations, that the avoidance reading is influential and unproven, and that a dreamer who finds the Jungian reading illuminating is doing something worthwhile that is not the same as discovering what the dream means.