Can we consider insulin receptor an enzyme? In other words, does the insulin receptor have enzymatic characteristics?
Yes, the insulin receptor can be viewed as an enzyme, since it catalyzes a reaction -- phosphorylation of tyrosine residues on its substrates. But since the protein has multiple functions, it's probably better to say that the insulin receptor "has enzymatic activity", rather than "is an enzyme".
The notion "enzyme" is not restricted to catalysis of reactions involving small molecules, like those in central metabolism. The IUPAC definition of an enzyme reads: "Macromolecules, mostly of protein nature, that function as (bio)catalysts by increasing the reaction rates". So "enzyme" is a broad term, including catalysis of all chemical reactions. DNA replication, mRNA transcription, protein synthesis and posttranslational modifications all involve enzymes. Those enzymes that act on small metabolites can be called "metabolic enzymes" to clarify their function.