Shinto IS a religious belief. It is a form of ancestor worship. Their attitudes towards war can be compared to the fictional Klingon philosophy; "To die in battle is to die with honor. To be defeated in battle is to die without honor."
In point of fact, the kamikazes died in honor in accordance to this philosophy. If they had not fulfilled their mission then suicide using a Seppuku was the only way to regain even the slightest bit of honor, for themselves, their ancestors and their Emeror. Not unlike the Klingon use of the Hegh'bat for the same reasons. So no, kamikazes were not performing suicide, they were fulfilling their duty for the honor of all.
Not unlike the young men first to storm the beaches of France on D-Day. They were not committing suicide. They too were performing their duty with honor.