Do the chain of command have the authority to make a soldier take an APFT every month if their score is not 250 and above?

Yes the CoC because because it's training , and as long as its on the training schedule then you are doing it as training.

No. The regulation is very clear. Twice a calendar year for Active Duty and AGR. Once a calendar year for Reserve.
Failures mat be tested at greater intervals.

reference this

A diagnostic APFT, yes. If you are familiar with FM 7-22 (particularly part 2, Strategy), it states that the APFT should be used as an indicator of how effective the PRT program is.
If a Soldier is falling short of the Unit goal, taking additional APFT's shouldn't be the plan to help them achieve the standard. Adapting the PRT program as a whole in order to improve the areas where the Soldier is falling short should be the long term goal.

The Commander can also require a Soldier to be taped, even if they are within weight screening guidelines... (AR 600-9 Chapter 3-2 e)

Ha! In my day, sometimes 1 a week/while in Garrison, & the Unit standard was 270. But, getting a 270 doesn't mean you can take 110# ruck, full gear, & weapon system & march 12 hours straight. It(score) was just so you could pass Promo's & enter schools, which never came since we spent 90% of our time in the field/deployed.

So, having a high APFT was pretty much mout for 95% of the unit.

I want to know why?

SFC (Join to see) Just a heads up. "Do" is not "Does", but it needs to be.