"The Fish" is a poem overflowing with imagery. This particular line, of course, comes right at the end of the poem, right before the fish is released. I tend to think the line "everything was rainbow, rainbow, rainbow!" has to be abstract.
To read it as a concrete image is to imply that all the narrator saw was rainbow on the fish or in the sky (for which there is no evidence). We know there is an actual reflected rainbow in the oil sitting in the bottom of the boat a few lines before this one; it seems unlikely the line refers either to that or the suddenly overpowering striping of the rainbow trout (which is, in reality, less a rainbow than just a red strip of color).
To read it as a figurative image implies all the symbolism and meaning traditionally attributed to a rainbow: beauty, hope, dreams, promise. These seem much more likely to be the view of a narrator who appreciates the character and perseverence of a fish that his battled its way through life and the intention of release than the more literal rainbow stripes.
The title tells us this is not just any fish ("The Fish"), and the opening lines tell us:
I caught a tremendous fish
and held him beside the boat
half out of water, with my hook
fast in a corner of his mouth.
Our narrator was not initially struck with its colorfulness; in fact, the description is of "brown skin hung in strips like ancient wallpaper." There is no literal rainbow to be seen on this battered and scarred yet beautiful fish--
--until everything was
rainbow, rainbow, rainbow!
And I let the fish go.
The rainbow seems so symbolic of the respect our narrator has for this creature (which shows itself in this new perspective on a battered, ugly fish) as well as the hope for the fish's future.