Molayne calls Kukui the Masked Royal twice. These are quotes from Molayne said directly to the Professor:
"You're still as fired up as ever, eh, 'Royal'?"
"But I was the captain once, my masked friend."
However Kukui never confirms this himself.
Whilst not as strong evidence, Molayne also asks you to return the Masked Royal mask to Kukui.
"And if we're done here, ... Could I ask you to do a favor for me? Professor Kukui forgot this."
One could potentially say that Moylane made a claim that Professor Kukui was the Masked Royal, but the claim was false. For all we know, he could have an identical twin not formally introduced in-game that is actually the Masked Royal, and after you give Kukui the mask, he was going to return the mask to his twin. (Whilst plausible, this is as unlikely as it sounds!)
So as you said, whilst it's clear that Kukui is the Masked Royal, it's not explicitly stated.
EDIT: In the anime, but not the games, it is clearly confirmed that Kukui is the Masked Royal. In episode SM140, the mask falls off the Masked Royal and reveals his true identity as Kukui to Ash. This knowledge is then made public in the next episode.