Here "equality" refers to "equal status" or "equal honor." Holding onto equality with God represents demanding that he continue to be honored as God is honored. Christ did not do that. Though he did not cease to be God, he ceased to act as God. AT: "he did not think that he had to have the same status as God" (See: [[rc://en/ta/man/translate/figs-metaphor]])
Paul speaks of Christ as if he were a container in order to say that Christ refused to act with his divine powers during his ministry on earth. (See: [[rc://en/ta/man/translate/figs-metaphor]])
Paul speaks of death here in a figurative way. The translator can understand "to the point of death" either as a metaphor of location (Christ went all the way to death) or as a metaphor of time (Christ was obedient even until the time that he died). (See: [[rc://en/ta/man/translate/figs-metaphor]])