We've implemented the $match operation for patient that takes FHIR parameters with the search criteria. How should this search work when the patient resource in the parameters contains multiple given names? We don't see anything in FHIR that speaks to this. Our best guess is that we treat it as an OR when trying to match on given names in our system.
We do see that composite parameters can be used in the query string as AND or OR, but not sure how this equates when using the $match operation.
$match is intrinsically a 'fuzzy' search. Different servers will implement it differently. Many will allow for alternate spellings, common short names (e.g. 'Dick' for 'Richard'), etc. They may also allow for transposition of month and day and all sorts of similar data entry errors. The 'closeness' of the match is reflected in the score the match is given. It's entirely possible get back a match candidate that doesn't match any of the given names exactly if the score on other elements is high enough.