What does my coworker mean that our company doesn't support "Comp"?

"Comp time" is time off in compensation for extra hours worked. For example, you had to work 60 hours last week to get a project finished, so you get to take Friday off this week. (I never said it was fair or equable). It is common in Information Technology, where there are requirements for sprints and/or off-hour support.

"Comp time" is compensatory time, as explained in wikipedia (and in many other online sources, if you google for compensatory time). The wikipedia entry says: "Time off in lieu; compensatory time; or comp time refers to a type of work schedule arrangement that allows (or requires) workers to take time off instead of, or in addition to, receiving overtime pay." It then goes on to state it is "legal in the public sector but not in the private sector". For example, the U.S Office of Personnel Management spells out in excruciating detail the comp-time rules for public sector workers. The terms "non-exempt" and "exempt" refer respectively to workers subject to, or not subject to, U.S. wage and hour laws. Typically, private sector hourly workers are non-exempt and required to take higher-rate overtime pay but not officially allowed to take comp time, while salaried workers are exempt, with comp time but not overtime pay available to them.