Only when the caster is below 5th level
The errata for the PHB has clarified the restriction further from the original printing:
To be eligible for Twinned Spell, a spell must be incapable of targeting more than one creature at the spell’s current level.
By default, eldritch blast does not have a range of self and is capable of targeting only one creature. It is thus eligible to be twinned.
However, the spell becomes capable of targeting more creatures once the caster reaches level 5:
The spell creates more than one beam when you reach higher levels: two beams at 5th level, three beams at 11th level, and four beams at 17th level. You can direct the beams at the same target or at different ones. Make a separate attack roll for each beam.
Thus, once the caster reaches 5th level and above, they can no longer twin eldritch blast.
Note that eldritch blast (and all other similar cantrips) scale with character level not class level as confirmed (unofficially) by Jeremy Crawford on Twitter. (See Do Cantrips use your character level or class level? for more discussion about that)
You seem to have misunderstood Twinned Spell slightly.
It doesn't cause a second instance of the spell to target another viable creature within range; it causes the same instance of the spell to target two viable creatures within range instead of just one. The relevant section of Twinned Spell reads as follows (emphasis mine):
... you can spend a number of sorcery points equal to the spell's level to target a second creature in range with the same spell.
This has a couple of implications.
First off, for spells like Polymorph, both of the creatures you're targeting turn into the same creature. However, you can keep concentration on both creatures, as it is only one instance of the spell, as opposed to two.
Secondly, for spells with a damage roll (like Chromatic Orb), you roll the attack roll separately for each creature, but you only roll damage once.
For spells that require concentration, both creatures are affected by the concentration effect. For example, if you cast Twinned Witch Bolt, if you hit both targets, you deal 1d12 lightning damage to them on the first turn, and you can use your action to deal 1d12 additional damage on each of your turns to each of the targets. If you fail a concentration check, you lose concentration on the spell for both creatures, not just one of them. Whether moving outside the range of Witch Bolt ends the spell for one or both creatures is up for interpretation by your DM (a strict reading of the RAW would suggest it ends for both of them, but I would personally rule that it only ends for the creature moving out of range).
In short, your polymorph spell turns the two targets into the same kind of creature.
Best Answer
No, the errata specifies that spells capable of targetting more than one target do not qualify for twinned spell.
From the errata:
Since part one of the spell can target up to 8 creatures, it's disqualified from being twinned. The option of targetting more is what disqualifies it, which is the same reason you can't twin magic missile if you only target one creature, or eldritch blast when it hits level 5.
Frankly I'd houserule yes if you were just using the 2nd part. But then again I'd also houserule using Magic Missile against one target to be able to be twinned, as well as any other spell that's only hitting one thing. But that's not what the rules say, that's DM privilege.