What type of responsibility obliges a soldier to exercise reasonable and prudent actions to properly use, care for, and safeguard all government property in his physical possession?

The answer is Custodial responsibility

Custodial responsibility refers the obligation to provide a certain protective supervision or guidance upon a certain object or another person.

This type of responsibility is imposed to any persons who are using property that we're not personally own. (if the property is broken under our care, we may be required to take care of the repair or replacement)