Overall, membranes seem to have a great deal in common, but on closer inspection it is revealed that membranes of different cells have unique properties. what is the primary component of membranes that gives membranes cell-specific properties?

Proteins.

Explanation:

Proteins are organic molecules made of amino acids which are arranged in a linear chain and joined together through a peptide bond.

Membrane proteins are classified depending on their location:

Integral membrane proteins are part of a cell membrane and can either penetrate the membrane or associate with one side of a membrane. Peripheral membrane proteins are transiently associated with the cell membrane.  

They perform many important functions:

Membrane receptor proteins: relay signals between the internal and external environments. Transport proteins: move molecules and ions across the membrane. Membrane enzymes: transform a molecule into another formCell adhesion molecules: allow cells to identify each other.

Self-reports would be the answer.