A khopesh or saber is also shaped like a scimitar. Are those swords also improvised weapons and not actual weapons?
If the flame blade is an object, then it is a sword of some sort given its shape. That would make it a sword weapon, even if it isn't on the non-exhaustive table of PHB weapons.
If it is not an object like the text implies by not doing any physical damage at all and only doing fire damage, then it can't be an improvised weapon.
Either way it can't be an improvised weapon unless used to pommel strike(if you rule it an object) or some other atypical use of a sword.