Most probably, this starts with a an untrue premise:
There are not so many laws about eye injury.
How many "laws" are there in the Hammurabi Code? — 282.
How many "laws" are there concerning consequences of eye injury?
One law for harm done by everyday accidents, brawls, malicious intent,
one for professional medical health & safety.
The "why" and "how" are much less important than "who does what to whom?".
A seemingly slightly higher number is only arrived at if we also count sub-sections.
Let's count them.
Accidental Injury (L196-201)
(196) If a man has destroyed the sight of another similar person, they shall destroy his sight.
(197) If he has broken another man's bone, they shall break one of his bones.
(198) If he has destroyed the sight of a working man or broken a bone of a working man, he shall pay one mana of silver.
(199) If he has destroyed the sight of another man's slave or broken a bone of another man's slave, he shall pay half his value in silver.
(200) If a man has knocked out the tooth of a man who is his colleague, they shall knock out his tooth.
(201) If he has knocked out the tooth of a working man, he shall pay a third of a mana of silver.
On first sight, that isn't "many". That is also evident in how the question presents its own numbers:
Q and putting out eye five times (196, 198, 199, 220, 247)
That is an entirely different class of "laws":
Medical and Veterinary Help (L215-25)
(215) If a physician has made a deep incision with a surgeon's knife on a man and has saved the man's life, or has opened out a man's eye- socket and saved the man's sight, he shall receive ten shekels of silver.
(216) If it was a working man's son, he shall receive five shekels of silver.
(217) If it was a slave, the slave's owner shall give the physician two shekels of silver.
(218) If a physician has made a deep incision with a surgeon's knife on a man and has caused the man's death, or has opened out a man's eye-socket and destroyed the man's sight, they shall cut off his hand.
(219) If a physician has made a deep incision with a surgeon's knife on a working man's slave and has caused his death, he shall make recompense with slave for slave.
(220) If he has opened up his eye-socket and destroyed his sight, he shall pay half of his value in silver.
In casuistic law (to regulate talionis), you cannot list everything.
Thus, you need example cases, here presented in a hierarchy: kill (very bad), eye-loss (permanent), broken bone (will heal) [only then how to apply it to classes (free, slave, animal)?]. We need to treat the hierarchy with salt, of course, as eg the magic spells part breaks our understanding much more than financial equivalents?
The question asks for laws about "eye injury". We see 3 equal cases of everyday damage, to be judged just sorted by class of people involved. For the actual injury, this is just one type of injury, "loss of sight". Thus we have basically one law number and additional subsections.
Then quality insurance to prevent or compensate iatrogenic damages. One that starts with operating around the eye and saving it to be compensated! Differentiating for carelessness and a codified risk/benefit ratio.
In relation to the hierarchy, the order of these "laws" present, it seems quite strange to count "eye injury" as "often":
L1–5 Unsubstantiated allegations
L42–56 Agricultural tenants
L67–69 Houses (sorting issues of sources)
L70–107 The merchant (–"–)
L108–111 Selling alcohol
L112–126 Other people's property
L127–195 Women and Children
L196–252 Compensation and fees
L253–258 Casual labour
L259–260 Fines for stolen implements
L261–267 Herdsmen and shepherds
L268–273 Hire of animals and equipment
L274 Tradesman's rates
L275–277 Rates for boats
L278-82 Slaves unable to work
— M.E.J. Richardson: "Hammurabi's Laws. Text, Translation and Glossary", T&T Clark International: London, New York, 2004.
So in reality, and taking the code for an actual compilation of actual laws (that is actually disputed), we do not see a very specific treatment of just one very specific injury. But a case of symbolic justice, standing in pars pro toto for a lot of similar cases.
The law of symmetry, 'essentially the same' bads be punished with an 'essentially the same' kind of bad whether actual 'eye for eye' or monetary. And a qualification for 'essentially not the same' treateed 'essentially differently'.
Hopefully, this kind of literal reading is partly dispelled with the alternative translation of "eye" used here (meaning often just sight, in many instances throughout the code very 'poetically'/metaphorically (('pleased his eye'/'his eye apple'))
"The eye" may have been 'very significant' in Mesopotamian culture. But the code of Hammurabi itself does only lend to significant insights towards medical practices around the eye being widespread. Just not: that 'eyes' were of that much more importance as alluded to in the question's various hypothesis.
The first qualification: It doesn't matter whether you kill by decapitating or breaking a neck, strangling or poisoning (or magic spells). What matters is that there is a dead body and 'someone did it'. What matters for the punishment is further the class relation between killer and victim.
The second qualification is related to intent: whether a doctor does harm by gross negligence/malpractice is treated differently compared to 'acceptable losses'?