Are multi-chain proteins synthesized as one biological unit?
Are multi-chain proteins synthesized as one biological unit? 


Sometimes yes but mostly not.



Some proteins are synthesized as one long polypeptide pre-protein which is cleaved by some proteases to yield multiple chains. After cleavage the intramolecular interactions become inter-molecular or inter-chain interactions. Insulin is a good example of this (See the figure below).






In other cases the chains are synthesized as different polypeptides which (with or without processing) subsequently associate with each other. In case of homo-multimeric proteins, the polypeptides are made by the same mRNA.