Is a linear function when y=x?

No. It's ANY equation that when you put it on a graph, the graph is a straight line.

That almost always means

Y = Ax + B or Ax + By = C

with A, B, and C, being any numbers.

Another thing it means is that the x and the y are never squared or any other power ... only the 1st power ... and they're never in a denominator of a fraction.