Wiring – Why are the bathroom light and fan wired to the same switch?

The answer to your only stated question is YES. To independently switch 2 different things you need to have 2 switched hot wires, you would have to run an additional wire from the wall switch location to the ceiling location.

The statement that "the light only has 14/2 running to the switch" tells us that one of those wires is a hot lead to the switch, the other is the switched hot that runs back up to the light (unless there are other/more wires that you did not mention).

It is possible that the two black wires (other than that going to the switch) are going separately to the fan and light. If so, the change is easy.

To test this do the following:

1. turn off the breaker
2. separate the three black wires nutted together
3. wirenut the black to the switch with one of the others and insulate the other in a separate wirenut
4. turn the breaker back on test the switch. Hopefully, the switch will operate only one of the fan or light.
5. turn the breaker back off
6. switch the black wires so the other one is connected to the wire to the switch
7. turn the breaker back on and test the switch. Hopefully, it will only control the other of the light or fan
8. Turn the breaker back off and optionally restore the original wiring

If the test results followed what I marked "hopefully", you are in business. Remove the other wire to the switch and wirenut it to two wires to run to one side of the dual switch (if one side of the switch has a removable bridge, just connect the single wire to that side). Connect the two blacks originally wirenutted together to the other side of the switches (if that side has a removable bridge, remove it).

For the pigtail wires, you should use the same guage wire as the rest. Examine the wires for printing on the insulation (look for an AWG number).

I'd also recommend against using the back push-in connectors. It's better to use the screw terminals.