Its always better to keep motors and digital ICs on separate supplies.
Motors and other inductive loads can generate large noise spikes.
The ULN2003 does have free-wheel diodes which are essential to prevent
circuit damage - the COM pin in that diagram must be connected to the motor
positive supply as shown.
Given that, and given adequate decoupling for the motor, sharing the 5V
rail may be OK - that motor takes a relatively low current and shouldn't
overload the Arduino regulator.
I'd suggest adding perhaps 100uF electrolytic across the GND and COM
terminals of the ULN2003 - remember polarity matters, GND is negative,
COM is positive here. This soaks up the current switching spikes at source.