Is Ash's explanation for the origins of his name correct?
Yes, this quoted dialog response by Ash is incorrect.

Aslan is not even a Hebrew word, it is a Turkish word, and it does not mean "daybreak" in Hebrew. Daybreak can be "shachar" (שחר) as you found, which is a very common name, or "zricha" (זריחה) which is the action of the sun arriving, and isn't used as a name.

Aslan, in Turkish, means a lion, and it is used a lot for names, even in Hebrew (not very common, but we do have this name as last name), this might be the source of confusion for that author.