Is y = 5 a linear function?

Yes it is a horizontal line 5 units above the x-axis. You know that it is horizontal because there is no slope or variable x. This line is a function just as any other line with the only "peculiarity" being that it is horizontal and thus has no slope.

Now a vertical line like x=3 is a line but not a function because it would fail the vertical line test ...

No, because the form for a linear function is y=mx+b; y=5 is just a line.