Does the "polymer" in CFRP directly and unambiguously imply the use of an epoxy resin?
You could use other polymers. Polyesters, vinylesters, and epoxies are all common in fibre-polymer composites and could be used. You can also use other special stuff if you need, eg, fire performance.
Epoxies are common because if you're using carbon fibres (rather than a variety of glass) you are evidently looking for a high performance component. Of the commonly used polymers, epoxy is the family at the high performance end of the range. Epoxies typically out-perform other resins both in mechanical properties terms and in environmental degradation.