Personally, I do not think you can use Twin spell because Protective link has effects that apply to both the affected creature and the caster of the spell. From the description of the spell (in the basic rules and online here):
As long as the target is within 20 yards of you, it gains a +1 bonus to AC and save rolls, and resists all damage. In addition, each time you take damage, you suffer the same damage.
The twin spell can only affect spells that target a single creature. Among the features of the Sorcerer class (in the basic rules and here):
To be eligible, a spell must be unable to target more than one creature at the current level of the spell.
Usually, for this rule, the word "target" should be interpreted broadly to mean "has an effect on". So, for example, a Fireball can not be paired because even if you only choose one target (a point in space), it does not affect any creature in the effect of its explosion. Fireball is also ineligible because the chosen target is a point in space and not a creature, but even a spell of the kind of Ice knife (which targets a single creature but then has other effects in an area) is not good.
Alas, the situation in this case is a bit more ambiguous, as decision tweeted by the creator of the game, Jeremy Crawford for a related question suggests that the caster of Protective link should not count as a target:
@ JeremyECrawford Warding Bond can he have several active throws if the carpets are filled? 3 sets of divided rings = 3 distinct periods in progress?
Jeremy Crawford @JeremyECrawford
You can maintain the protection link on multiple creatures at once if you have a pair of rings for each launch. #DnD
The normal reason for waiting for you to not be able to launch Protective link many times if the thrower was a target, it is that there is a clause in the spelling rules that says when it ends early:
It also ends if the spell is cast again on one of the connected creatures.
So, if the caster was a target, it would end the previous cast by casting the spell because they are the target of both spells.
Although Jeremy Crawford's tweets are not official decisions nowadays, given his interpretation, I'd expect him to say Twin spell should work too. Who knows how the hardware components should work in this case, since the launcher and the willing target must wear a ring for the duration of the process. Maybe you need three or four rings (the caster may have to wear two)?
It's a bit of a mess. Personally, if I were GM and had to interpret these rules for my own table, I would not allow Protective link to be twinned. You will probably have to ask your GM for a decision for your game.