Straight from the DMG Errata:

Attunement (p. 136)

The first paragraph ends with a new sentence: “If the prerequisite is to be a spellcaster, a creature qualifies if it can cast at least one spell using its traits or features, not using a magic item or the like.”

emphasis added

Therefore, for the purpose of attunement:

If a creature is able to cast a spell by virtue of their own ability (not from an item), whether that be race, class, or feature, you are a spellcaster.

Flame Blade cannot be used with Booming Blade

Although diegetically the Flame Blade spell gives you a flaming weapon, you cannot use it for ordinary weapon attacks like what Booming Blade requires. The Flame Blade spell only gives you the ability to make Melee Spell Attacks with your action, dealing the amount of damage specified by the spell.

You evoke a fiery blade in your free hand. The blade is similar in size and shape to a scimitar, and it lasts for the duration. If you let go of the blade, it disappears, but you can evoke the blade again as a bonus action.

You can use your action to make a melee spell attack with the fiery blade. On a hit, the target takes 3d6 fire damage.

—Flame Blade, Player's Handbook, pg. 242

This is in contrast to a spell like Shadow Blade, which does yield a weapon that can then be used with Booming Blade or Green Flame Blade, because the weapon provided by that spell is, for all mechanical purposes, a proper weapon.

You weave together threads of shadow to create a sword of solidified gloom in your hand. This magic sword lasts until the spell ends. It counts as a simple melee weapon with which you are proficient. It deals 2d8 psychic damage on a hit and has the finesse, light, and thrown properties (range 20/60).

—Shadow Blade, Xanathar's Guide to Everything, pg. 164

"But what if my DM is super nice to me and lets me do this?"

If it were possible to use Flame Blade with Booming Blade, as a consequence of DM houserules, you would make a Melee Spell Attack using the Spellcasting ability associated with Flame Blade, which as a Druid spell, is Wisdom, and then add the Booming Blade effects on top of the Flame Blade damage. This is because Booming Blade calls for the user to make a normal melee attack with their held weapon, and uses all the normal rules for an attack made with that weapon. Since Flame Blade's action uses a Melee Spell Attack, you'd have to translate that to its use with Booming Blade.

Conversely, if you were to use a spell like Green Flame Blade, obtained as a Wizard spell, with your Flame Blade, you'd have the strange (but mechanically solvent) situation where Wisdom is used as the Ability modifier for the Attack Roll, and then Intelligence is used as the Damage Bonus on the secondary effect of Green Flame Blade.