I believe the myc tag is mainly used for immunodetection and immunoprecipitation, whereas the His tag is mainly used for purification by IMAC chromatography. (Anti-His-tag antibodies are available for detection as well, but they don't always give satisfactory results. In principle, the myc tag could also be used for purification, although it would only be practical on a very small scale.) Having both tags gives you both a good method for detection and a good method for purification.
The rationale for including both a C-Myc tag and a 6xHis tag in the same plasmid is to provide flexibility and redundancy for protein detection and purification. The C-Myc tag allows for easy detection using anti-C-Myc antibodies, which is useful for Western blotting, immunoprecipitation, and immunofluorescence. The 6xHis tag facilitates purification using nickel affinity chromatography and can also be used for detection with anti-His antibodies.
Yes, you can use only the 6xHis tag for mammalian expression if your primary goal is protein purification. However, having both tags provides additional options for detection and purification.