Objects with no name, like "the Sun"
Solution 1:
Posted as answer, as requested:
I think that 'the sun' counts as a name. There's only one; we refer to other giant, bright balls of hydrogen as 'stars'. In fact, you capitalize Sun if you're referring to it in an astronomical context
Solution 2:
Your question contains an implicit assumption which is unfounded: the words sol in Latin, helios in Greek, and güneş simply mean "sun". They are names for the sun in exactly the same way that the English word is the name for the sun. If you doubt this, consider the following: is there some generic word for "sun" in either Latin or Greek that contrasts with the proper names sol/helios? (I don't know Turkish, so I won't comment on güneş.)
In other words, we do have names for the big bright light that appears during the day, and the smaller, waxing and waning disk that appears at night. Those names are "sun" and "moon". Other burning balls of hydrogen are called "stars", and satellites orbiting other planets may be "moons", but they're not the moon.
(HT: Nathan, who gave the correct answer in a comment.)
Solution 3:
"The sun" is its name in English, just like "Sola"/"Solen" (definite form of "sol") is its name in Norwegian. While "sun" may refer to the hydrogen fusion ball in the sky or its light, just like "sol" does in Norwegian. A name with "the" or a definite article doesn't make it less a name. If a name for something is in a definite form (i.e. "the" in english), it usually implies it's the only one.
For instance "united states" could be any union of states, while "The United States" is almost definitely a short form of "The United States of America" aka. "USA".