I generally agree with this answer that for clarity and consistency it'd be best to avoid multiple ids pointing to the same entity.
Sometimes however, such a situation arises naturally. An example I work with is Polish companies, which can be identified by their tax id ('NIP' number) or by their national business registry id ('KRS' number).
In such case, I think one should first add the secondary id as a criterion to the search endpoint. Thus users will be able to "translate" between secondary id and primary id.
However, if users still keep insisting on being able to retrieve an entity directly by the secondary id (as we experienced), one other possibility is to provide a "secret" URL, not described in the documentation, performing such an operation. This can be given to users who made the effort to ask for it, and the potential ambiguity and confusion is then on them, if they decide to use it, not on everyone reading the documentation.
In terms of ambiguity and confusion for the API maintainer, I think this can be kept reasonably minimal with a helper function to immediately detect and translate the secondary id to primary id at the beginning of each relevant API endpoint.
It obviously matters much less than normal what scheme is chosen for the secret URL.