Yes, on any RDBMS, if you still have a unique index with non-nullable columns to identify individual rows.
The difference between a "unique key" and a "primary key" is a PK does not allow NULLs. NULL does not compare to NULL so you can not uniquely identify a row with a nullable unique key. Especially if your RDBMS ignores NULLs for uniqueness (e.g. SQL Server allows at most one NULL in a unique index)
Any of the indexes (A PK is an index) can be clustered: but every table should have clustered index (outside of staging table that are cleared after use). The clustered index can be the PK, the unique key, or another index.
I'm sure MySQL allows separation of index uniqueness and clustering