mRNA and Protein relation
1) Can two different proteins have the same mRNA?

This isn't clear enough to give you an answer:


Can two different proteins be made from the same mRNA molecule? 


Yes, polycistronic mRNAs can encode several proteins which are translated separately (we can engineer this using internal ribosome entry sites, IRES). 
Yes, different proteins can be translated from one open reading frame by failure of peptide bonding between two amino acids (we can engineer this using 2A peptide sites). 
Yes, identical proteins can be translated from one mRNA and then modified differently after translation to yield different "final" proteins.
Yes, identical polypeptides can be translated from one mRNA and then incorporated in different protein complexes.

Can two different proteins be made from the same mRNA code?


No, not made in the sense of translated. 
Yes, as above, identical proteins/polypeptides can be made and treated differently after translation.



2) Is it possible that the gene types which encode the synthase are the same?

Gene types? Do you mean genotype, i.e. allele? What synthase? Sounds like a homework question to me.