Voltage to GPIO pins
When we set the bit to 1 in the data register, corresponding to that output pin, does it mean that the microcontroller starts providing constant voltage to that pin

Yes.¹
Actually, the start the device doing that is when you write the direction register – at that moment, what's currently in the data register becomes applied to the pin.

If so, is the voltage provided through the data bus to the data register (which I don't think it to be the case) or is there a wire from the pin to power source to get the voltage?

ENODEF; that's not defined, it depends on how your microcontroller is built.
In most modern microcontrollers, there's specific power rails that supply mainly the IO pins' out- and input circuitry, and the digital logic controls that circuitry. But, especially in older processors, these pins often weren't strongly driven (able to supply a few microampere only) and that might have happened from some internal logic.

¹ &quot;Constant voltage&quot; is a bit of a simplification: obviously, microcontroller pins can't source or sink infinite current, so, there's no &quot;perfect&quot; constant voltage there; also, there's different kinds of IO types in microcontrollers: BJT or CMOS push/pull outputs, open drain, open collector, differential outputs, specific signalling levels, there's often things like optional slew rate limiters...