DisplayName is not guaranteed to be unique, so you might get unpredictable or multiple results from a search on DisplayName. DisplayName is often the kind of field I would expect to search with a wildcard (contains, starts with, etc.) just because of the vagaries of how those are generated and updated.
All of the fields that https://learn.microsoft.com/en-us/powershell/module/exchange/get-recipientpermission?view=exchange-ps searches on for Identity or Trustee are fields that are unique.
For identity:
- Name
- Alias
- Distinguished name (DN)
- Canonical DN
- Email address
- GUID
For Trustee:
- Name
- Alias
- Distinguished name (DN)
- Canonical DN
- Domain\Username
- Email address
- GUID
- LegacyExchangeDN
- SamAccountName
- User ID or user principal name (UPN)