Using Gedmatch's file diagnostics, an autosomal DNA test kit I manage shows that there are 31200 (4.4%) no-calls. In most matching algorithms, a no-call is treated equivalently to a match. Therefore, as Gedmatch states:
This kit has an unusually high number of no-calls, which usually results in a larger number of false matches.
Which would explain why there are so many matches for this kit with (likely) no actual genetic relationship.
This kit was tested through FamilyTreeDNA.
Is this truly an unusual margin of error? Is it unreasonable to ask FamilyTreeDNA to retest on a new DNA sample?