It looks like the weapon display in your equipment page does not correctly display weapon damage while in animal form. Did some tests by slashing away at stuff in various forms and looking at the range of damage numbers I was getting in the combat log. This was at level 7 using one-handed flame blades with a WIS bonus of +5 and an STR bonus of +4. The numbers below are with no buffs, and ignoring attacks that did criticals or grazing blows.
1) Dwarf form, 1-handed flame blade. For each non-critical attack I was getting a 1d6 fire attack (the flaming effect), followed by a second fire attack in the range 8..13, which looks like a 1d6+7. The +7 would be +5 from WIS and +2 from flame blade level. [Inventory says Damage: (3-8 ) Fire, Bonus: +12]
2) Bear form, 1-handed flame blade. Did a 1d6 fire attack (flaming) followed by slash attack in the range 7..14. (It's possible that the combat log only lists a single attack type, so it may be fire/slash/bludgeon in reality. Bears are supposed to do slash+bludgeon with their claws.) 7..14 is 1d8+6. The +6 (+2+4) suggests that it is using STR bonus for damage. Note that it is 1d8 in bear, not 1d6. [Inventory says Damage: (3-8 ) Fire, Bonus: +14]
3) Wolf form, 1-handed flame blade. Did a 1d6 fire attack (flaming) following by a 7..12 (1d6+6) slash. Again, the +6 suggests +2 level and +4 STR. [Inventory says Damage: (3-8 ) Fire, Bonus: +14]
Second experiment, with +2 WIS item removed. WIS and STR bonuses are now equal at +4:
3) Wolf is still doing 7..12 on its slash attack. Not affect by change to WIS.
2) Bear is still doing 7..14 on its slash attack.
1) Dwarf now doing 7..12 on its second attack. This IS affected by WIS.
So, while in wolf form, your weapon is always a 1d6 STR-based claw; in bear form it is always 1d8 STR-based. Only in dwarf form does flame-blade become WIS-based.
Did some further tests on critical threat range. For bear with 1-handed flame blade, DOES NOT crit on 18 or 19, only on 20. Multiplier looks to be x2, but it was so rare it's hard to be sure.
Wolf crits on 19-20 but NOT 18. Multiplier looks strange, the calculation doesn't seem to be exact, but I'm guessing at x2.
Attack bonuses were all over the place. This might be due to the different bonuses with each phase of the attack animation.
I didn't test dual-wield. Also didn't test Keen. It may or may not transfer across to claws. Not sure whether it would need to be Impact or Keen for that matter.
At level 7, all flame blade does (over being unarmed) is add a +2 enhancement bonus, the Flaming effect, and also adds Fire to the damage types.