The whip blade is unbalanced for dexterous characters and pointless for strong characters.
(I am assuming that the interaction to toggle forms is completely free because no action is mentioned in the whip blade's description)
Toggling forms
In most cases, a character can change their reach once during their turn by dropping a weapon and unsheathing another weapon that has more or less reach.
By comparison, the whip blade can toggle its form to change the reach as many times as necessary per turn without the hassle of dropping stuff onto the ground. I think that the toggle ability is decent in the hands of a tactical player, but not game changing.
Dexterous characters (dex ≥ str)
The whip form's lack of finesse is inconsequential because the whip form uses dex anyway.
The whip form is strictly better than whips because of the extra damage.
The whip form is at least on par with rapiers and scimitars. However, if the dexterous character has the dual wielder feat then the whip form is strictly better than scimitars.
Moreover, the whip form is strictly stronger than the blade form too, so the toggle ability is useless outside of some really rare scenarios where having reach is a negative.
Overall, I would almost always use the blade whip with a dexterous character over any other melee weapon.
The exception are rogues because they need the finesse property for sneak attacks. However, rogues are not proficient with whip blades, so it's not worth analyzing.
Strong characters (str > dex)
The finesse property is meaningless to strong characters and so the blade form is strictly worse than all one-handed martial melee weapons except for the whip and the lance, and generally worse than all martial melee weapons.
The whip form is strictly worse than whips when you consider the reduced chance to hit from using dex instead of str.
Overall, I would almost never use the whip blade with a strong character.
My recommendation
Just flavor a regular whip to be a whip blade. When the enemy is within 5' describe the attacks as belonging to a solid blade, else describe the attacks as belonging to a whip-like sword. Guaranteed balanced.
Once the game progresses to the point that magic weapons are available, put something like this in the loot.
Whip Blade
Weapon(whip), rare
You have a +1 bonus to attack and damage rolls made with this magic whip. When you make an attack with it during your turn, you can forgo its reach property to deal an extra 1d4 slashing damage.
This is a great start
I always love when player's have a idea for a unique ability they want for their character so even if this was slightly over-powered I would consider allowing it. Overall it looks pretty good and I think you've done a good job and thinking through the balance of the spell overall. I'll just highlight a few things that I noticed from reading it.
"Incandescent light" is ambiguous
While I thematically understand what you were going for. Emitting light is a mechanical term in D&D and "incandescent" isn't one of the types. It would be better to simply stick with "dim light" or "bright light" and specify a distance. Or if you don't intend for it to emit light, consider rewording or removing this entirely.
The area of effect is wrong
A 5 foot cone would only hit the first target. I think the correct area for the effect you want is a 10 foot cone originating from you and directed toward the target of the attack.
Damage is underwhelming
For a second level spell the damage isn't that spectacular. Of the spell you compare it too, Scorching Ray has significantly more range and Dragon's breathe has up to 10 activation per casting. I think you could consider increasing this, consider spells such as Burnings Hands and Ice Knife in your comparison.
No save for additional targets
Basically this is a AoE spell that only requires an attack roll against the closest enemy in the range. If used cleverly you could attack a low AC target and have a high AC target as one of the additional targets which would then take automatic damage. Instead I would consider something like the Ice Knife mechanic where the additional effect is trigger hit or miss but allows a saving throw for half damage.
Missing the attack roll is a big cost
Currently there is a huge opportunity cost to cast this spell. Assuming the fighter is at least level 5 they are giving up their extra attack and a second level spell slot for the chance to do some extra damage. If they miss the attack roll it is all wasted.
I would suggest either activating the effect hit or miss as mentioned above or looking at the Paladin's smite spells that activation on the next melee hit instead of as part of the action to cast it.
Conclusion
I like the spell overall, I think its a good start. And while I think it could be improved I don't think it is so unbalanced that I would ban it at my table. If you do make changes remember that each of the comments I made were to that issue in isolation. Some of the issues are related so if you modify one consider how it effects the others.
Good luck and remember to wait at least 3 days before posted an new iteration to give time for multiple answer to come in.
Best Answer
The attack needs to be mandatory & immediate.
"As part of the same action, you can make a melee or ranged attack..."
So if I choose not to make that attack, the transformed "ally" remains a weapon? New plan: cast charm person on my enemy, convince them to help me fight someone, cast this spell on them, store the transformed weapon in a box, bury the box in the woods, and move on with my life.
Phrasing Suggestions
It feels a little weak if used normally.
This spell has 2 core effects: it lets you make an attack for 1d8 damage with a stat that probably isn't your best one, and it teleports an adjacent ally to the target of your attack.
The teleport effect is potentially nice, but has enough limitations (ally has to be adjacent to you, range limit, not actually a teleport) that it doesn't pull its weight on its own as a level 1 spell.
The damage, in turn, is almost the bare minimum possible. Sure, it's not a d4 or a d6, but the use of Str or Dex rather than your casting stat means you're less likely to hit and get less static damage.
For comparison, consider the 1st level spell catapult from Xanathar's Guide to Everything. It launches an object of your choice (with weight limits) within 60 ft of you up to 90 ft at a target, who must make a Dex save or take 3d8 damage. Compared to that spell, the only thing friend blade has going for it is saving your ally a round of walking.
Remember, any round you cast this spell is a round you didn't cast fire bolt for 1d10 (or more, at higher levels) damage with a 120 ft range.
I would recommend either looking to make this spell more reliable, or make it more of a high-risk/high-reward option (with the ally-teleport as a consolation prize if you miss). You have three dials you can turn to adjust these: whether the attack is a spell attack (using the caster's spellcasting attribute) or a weapon attack (using the better of Str/Dex), how much damage it deals, and what the thrown range is.
For example, you could leave it as a weapon attack, up the damage, to 2d12 or so, and give the thrown modifier a range of (15/60). That means the caster can potentially move their ally a good distance, with the added bonus of a low-but-not-zero chance of doing some good damage.
Alternately, you might change it to a spell attack, bump the damage to 1d10 or 2d6, and give the thrown modifier a range of (45/45). That gets you fire bolt damage, trading a bit of range (which isn't needed in most encounters) for a useful effect of moving your ally farther than they could likely go in one round without Dashing.