No, because Warding Bond is still cast on the caster despite them not being the target of the spell.
The core question we have to answer is "Does a spell being 'cast on' a creature mean the same as a spell targeting a creature?"
If the answer to that question is yes, then you can cast Warding Bond as many times as you have spell slots and pairs of platinum rings, as you are not the target of the spell when you cast it. If the answer is no, then you can't benefit from multiple castings, as you can have a spell 'cast on' you without being the target of the spell.
Per Blessed Healer (PHB page 60),
Beginning at 6th level, the healing spells you cast on others heal you as well. When you cast a spell of 1st level or higher that restores hit points to a creature other than you, you regain hit points equal to 2 + the spell's level.
This feature seems to be written specifically to include Mass Cure Wounds, which is the only Cleric spell that uses a point of origin rather than targeting a creature. Otherwise the class feature could have stated "when you target a creature other than yourself with a spell that restores hit points". Mass Cure Wounds, despite being a spell cast on a creature, is not targeting a creature. So the answer to "Does a spell being cast on a creature mean the same as a spell targeting a creature?" would appear to be no. Thus the caster of Warding Bond would count as having the spell "cast on" themselves, and would not be able to benefit from multiple castings.
Incidentally, we can also check to see if Warding Bond is a valid spell to be Twinned by a multiclassed sorcerer. Per Twinned Spell (PHB, page 103),
When you cast a spell that targets only one creature and doesn't have a range of self, 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. To be eligible, a spell must be incapable of targeting more than one creature at the spell's current level.
If you can cast Warding Bond as many times as you have spell slots and pairs of platinum rings, as you are not the target of the spell when you cast it, then Warding Bond is an eligible spell to be Twinned. By the contrapositive, if Warding Bond is not an eligible spell to be Twinned, then you can't benefit from multiple castings of Warding Bond.
This question has been asked before. According to Rykara's accepted answer, Warding Bond is not eligible to be Twinned. Therefore you cannot benefit from multiple castings of Warding Bond.
Per Sage Advice Compendium,
[NEW] Can my sorcerer use Twinned Spell to affect a particular spell? You can use Twinned Spell on a spell that …
- targets only one creature
- doesn’t have a range of self
- is incapable of targeting more than one creature at the spell’s current level
If you know this rule yet are still unsure whether a particular spell qualifies for Twinned Spell, consult with your DM, who has the final say. If the two of you are curious about our design intent, here is the list of things that disqualify a spell for us:
- The spell has a range of self.
- The spell can target an object.
- The spell allows you to choose more than one creature to be affected by it, particularly at the level you’re casting the spell. Some spells increase their number of potential targets when you cast them at a higher level.
- The spell can force more than one creature to make a saving throw before the spell’s duration expires.
- The spell lets you make a roll of any kind that can affect more than one creature before the spell’s duration expires.
Additionally, and I'm not positive this interview is accepted in answers on this site, from this interview starting at 28:03,
There are a few other types of spells that cause questions related to Twinned spells. ... The ones that are in a gray area are spells that have you choosing a creature, doing something to that creature, and then something else happens that might affect more creatures. Great examples of this sort of spell are ice knife and green flame blade. ... In each of those spells, you target a creature and you do something to them. ... Then green flame blade lets you pick a different creature to cause some additional damage to that creature, and then ice knife causes this explosion around the creature you hit with your spell attack. And so people ask, "Are those secondary creatures targets for the purposes of twinned spell?" The answer is yes they are.
These taken together would indicate that Warding Bond counts the caster as a "secondary creature target", and would therefore not be eligible to be Twinned. By the contrapositive, since Warding Bond is not an eligible spell to be Twinned, then you can't benefit from multiple castings of Warding Bond.