Can a roll of copper tape act like a solenoid when connected to electricity?
How does it compare to a solenoid of the same size made from copper wire?
Of course it can. There needs to be insulation between layers. Other than that, copper tape is just copper wire with high aspect ratio rectangular cross-section. Compared to small solenoids wound with thin wire, copper tape will have orders of magnitude lower resistance.
But you probably are asking in context of a particular application, so you won't get a particularly useful answer unless you get more specific.