Alright so first of all, good on you for not just trusting the wiki.
Secondly, here is a full (unofficial, of course) translation of the Super Bomberman 5 manual.
It specifically states that the game takes place in "alternate-dimensional world that 'Terrorin' has created" (p. 8). The wording here is 異次元世界, 異次元 meaning "different dimension" and 世界 meaning "world".
On page 15, regarding Terrorin, it says this: "Suddenly appearing from an other-dimensional space, he created Terrorin World, took the name of Emperor, and steeped himself in mystery." The specific wording here is actually 空間, meaning "space" as in "area" (not like outer space).
Unfortunately, that's all the information we get! The guidebook is a straightforward guide and has no lore information within. Not even a story manga!
I hope this helps~