If we were comparing competing similar texts from these periods that described the birth of some ancient king ...

A folklorist would say very quickly that all these reports about the birth of a culture hero were pure folklore with the possible exception of a single fact: a baby was born. That would be contestable if the existence of the king was also being contested, but let's say that historians could agree that this ancient king really existed. Given that assumption, it would be clear that all the stories and all the details associated with the birth of this king was nothing more than oral tradition that was drifting along until it attached itself to this culture hero.

The process that caused great men to attract stories of remarkable births was described as early as 1914 by Otto Rank; see his The Myth of the Birth of the Hero and other writings, ed. by Philip Freund, 1932 with numerous reprints.

So, how accurate is the Quran's account of the birth of Jesus compared to that of the Bible? A folklorists would answer that they have the same degree of accuracy; namely they both have no accuracy at all. That's what a folklorist would say if this were about an ancient king, but since it is about Jesus, a folklorist would receive a shit storm for saying that, so a folklorist would hem and haw and try not to get covered in shit by said storm. If a folklorist were to say that about Jesus, True Believers would be flushed out of every corner to point out that the Holy Gospels were written by people who knew Jesus and Mary and they received these accounts directly from them, so the Holy Gospels described that birth EXACTLY!!!! as it happened. If this were about an ancient king, the folklorist would respond that these sorts of legends attach themselves to culture heroes very quickly - sometimes even while the culture hero was alive. It is a natural process, and in fact it is a process that elevates that ancient king, because it indicates how clearly he was revered during his life and shortly after his death. But the folklorist would add that the details (or these type of details) are found attached to many other culture heroes, so it is clear that they are matters of legend rather than fact.

But this is Jesus, so I won't say any of that because I don't want to be in the midst of a shit storm.

edited to correct minor typo.