Just obtained Aloth in my party and not long after I was able to level him up, I noticed he came with a grimoire which shows 2 spells for each power levels 1 to 9. When I go to level up, the ability upgrade screen shows the spells he knows without the grimoire equipped as "known already" but not the 2 skills he knows in the grimoire. When the grimoire is equipped it allows me to upgrade to those skills even though he knows them already.

 

If I choose a skill in the grimoire to select, it slightly upgrades it's damage and shows the skill twice under the character statistics screen. Is this normal or a bug? I was quite confused when it came to leveling him up.

Well, since you can switch out Grimoires it only makes sense for the game to let you pick spells when leveling up regardless of what's in your Grimoire. Basically, spells you learn when leveling up are "permanent", and you get to supplement them with Grimoires.