This one seems pretty obvious in the picture for me. Amon's not allowed until you clear the first area since she's a story character that hasn't joined you yet. Go through the first few quests then try her out.
Oh... well it might be because you didnt unlock her first version yet. (Don't worry it will be quick)
edit : too slow
I just expected the added bonus to be displayed somewhere easier to notice like the skill button that displays what skills are activated or something but since it's not, unless you read the wiki or was trying to figure out the math behind total power, you probably wouldn't have ever noticed. It's not even mentioned in the How to Play tab unless I missed it.
It was actually vaguely alluded to in the in-game help, though it only mentions the specialized weapon bonus to souls (kamihime also have specialized weapons). The bonus itself is applied without any notification.
Here's the entry from the JP wiki.
And the relevant section of the in-game help. I don't see a listing of the specialized weapons for souls, but my guess is that the specialized weapon is the one their sprite uses in battle.
No, neither the weapon they wield nor the one they were freed from are certain indicators of which one is their favored weapon. The jp wiki displays them in list view and in each entry. dmm players on discord confirmed the weapon correction bonus, and reminisced about the nerd rage incurred by dev revisions.