Relationship with a dimension table that has several keys

Power BI doesn't have joins like SQL databases. The are filter relationships, and the DIM table must be unique. You may create a compound key in Power Query using someting like =[Column1] & [Column2] & [Column3] in both your FACT and DIM table and create the filter relationship on that. You can then hide both of those fields in the model so your user never sees them and can use the description or whatever they are used to seeing.