Does current draw of separate pins of a microcontroller affect each other?
This would happen if you limited the maximum current sourcing capability by putting a resistor in the Vdd line – or the maximum current sinking capability by putting a resistor in the GND line.

Don't do such nonsense. Put the current limiting resistors on the individual I/O pins as needed.

If your battery has a high internal resistance (for example, a button cell), this is the same as the nonsensical resistors in Vdd and GND line.