The question here, is very basic in nature but I really can't get the intuition behind it.
Consider the following translations:-
Degree swept by hour hand in 1 hr:-
$12 hr = 360^\circ$
$1 hr = \frac{360^\circ}{12hr} * 1hr = 30^\circ$
Degree swept by hour hand in 1 min:-
$1 min = 30^\circ * \frac{\frac{1}{60}min}{1 min} = 0.5^\circ$
In the 1st translation, 1hr is multiplied with the fraction while in the 2nd translation, 1 min divides the fraction.
Notice that even without using these 2 factors, there would be no change in the answer, more precisely, no change in the magnitude of the answer.
While using unitary method, multiplying or dividing with such single-valued factors is a normal step in evaluating the answer, often not shown to make steps less verbose.
But, my question is are these factors there only for to cancel out the like units so to bring out the required unit in the answer or is there any significance about these single-valued factors (like any information these factors are encapsulating)?
Also, if there isn't any significance about these factors, can it be said that they are just being used out of necessity to have a more complete consistent answer without any information that can be deduced from them?
I think you made an error.
Consider this: $\displaystyle\frac{360º}{12 \,\text{hours}}=\frac{30º}{1 \,\text{hour}}.$
Alternatively, you could try $360º \times \frac{1 \, \text{hour}}{12 \, \text{hour}}=30º$.
You'll cancel out the units or end up with the same unit combination on each side.
Even for the last conversion, $\displaystyle \frac{30º}{1 \,\text{hour}}\times\frac{1 \, \text{hour}}{60 \, \text{minutes}}= \frac{0.5º}{\text{minute}}$