Scifi has been enthusiastically using the name "Sol" to refer to our star for a long time. That's just taking the Roman name for the Sun in the same way the planets all follow a Latinate naming convention (except for Earth, which we'd call "Terra" if it did).
Science, or at least the IAU does not recognise the use of the name "Sol" but that hasn't stopped it's usage by some scientists (whose heads are likely full of those scifi stories!). Mission days on other planetary bodies like Mars are referred to as "Sols" for instance. Were it ever needed to adopt a specific individual name for the Sun I suspect Sol would be the chosen title without a second thought.
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Matt say: The Suns name is the Sun based on the latin word Sol. Other stars are not 'Suns' they are stars with names such as Polaris or Sirius. Just like in reality there are not Moons of other planets but natural satellites with names such as Titan, Ganymede etc. When these words were derived humans didnt know of other solar systems, galaxies or satelites of other planets. There was the sun and the moon.
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