[Thinlómien]

It's all logical, Legate, if you think of it. When Legolas says "Nay! Sauron does not use the Elf-runes", I think he simply means that Sauron does not use (a) elf-rune(s) as his symbol(s), but he does not mean Sauron wouldn't use them elsewhere (the ring etc.).
I wonder if the Ring's script being in tengwar has something to do with Eregion and gwaith-i-mírdain...?

And about the "international s-rune"... For Legolas, the rune was/seemed an Elvish rune, not any other system's rune, since he is an elf himself. It's like a today's western european or american or australian person would see the symbol "s" somewhere, you'd automatically think of the western letter "s", not some snake-shaped hieroglyph, that's happen to mean "s" too... Okay, that was a bad example, but I hope you got my point. And Aragorn (and Gimli) just did not see a need to comment it could be a rune of another system's, since that was of no importance right then. (Aragorn could maybe have even "recognised" the rune as elvish the way Legolas did, as he grew up in Rivendell among the Elves.)