ArcGIS joins support only one field, not multiple fields.
A workaround that people sometimes do is concatenate fields, that is create a string field in both tables that includes all the data, for example "123_4_5" and use that as a join field.
If the datasets are in a file geodatabase, you can do a more complex query using the Make Query Table tool. This is limited to only some data formats and familiarity with SQL is recommended. There are good example in the help fortunately. The general approach for this is to do a many to many join on one of the fields and then query for rows that match your multiple fields.
Make Query Table—Help | ArcGIS for Desktop