RAW
As the rule text you quoted says, a (singular) Dex save is rolled before determining which ray the subject is struck by. Thus even if a target is struck by two rays, they have already made their one save, and its success or failure is applied to both effects.
Narrative
The spell itself produces rays which fan out in a cone, some striking one target, others striking others, some missing targets altogether, as illustrated here (note this is an artist's interpretation of the 3.5e version of the spell, but thematically the spell has not changed between editions).
Dexterity saves typically represent your ability to quickly get out of the way of a harmful effect; because of the particular spatial pattern of the rays, it does seem odd in a narrative sense that you are either struck fully by two rays (single failed save) or mostly avoid both of them (single successful save). It feels odd that it is simply not possible RAW to partially avoid one ray and yet be struck fully by another.
At your table, you may find it more narratively satisfying to use the initial save for the first ray determined after the 8 is rolled, and then roll a second save for the second ray.
Choosing to play this way will make the spell outcome less swingy / more often closer to the average result, but will have a negligible effect on overall outcomes. The spell is 7th level, and the slot can't be restored by arcane recovery or purchased with sorcery points (probably for design reasons). From the time you get the spell at 13th level you will be casting it at most once per day until you are 20th level. Even if you always use prismatic spray for your once-per-day 7th level spell cast, the chance for any target to be struck by two rays is 1-in-8, and the chance for the two saves (if allowed) being the same is only slightly more than one half. Thus rolling the two saves separately is going to change the results for very few targets and will not come up very often. If your cone area is target-rich it is more likely of course, but that in itself is going to be a more rare situation.
Thus, if making this non-RAW change to the way the spell works adds to your player satisfaction, go ahead and do it.