If you listen to LDS teachers or read their doctrines you will see the very conscientiously worded phrase "the only God and his Son" rather than "God and his only Son." It's the same set of words but the dots are connected differently.
The difference seems to hang on the word "only". What does it connect to?
It appears that the operative phrase in John 3:16 is
τὸν υἱὸν τὸν μονογενῆ (
ton huion ton monogenē). Perhpas somebody with some real Greek knowledge could expound on this specific verse, but the the "only" bit seems to really indicate that he was an only son Son, not just by that method. In fact many translations1 don't even addd the word "begotten" although the original word is a compound that includes something about being a child, presumably in order to clarify in modern language what the adjective only is modifying. The meaning of the phrase in "King James English" which is retained in many English translations does not help us understand exactly what the verse is emphasizing. (See this answer for another example.)
John 3:16 (ESV)
For God so loved the world, that he gave his only Son, that whoever believes in him should not perish but have eternal life.
Some English translations go so far as to add extra emphasis to the fact that he is an only son by stating that bit twice without using the word 'begat'.
So it looks like we need to look farther than just this verse. So let's follow that word to some other usages in the Bible. In the following verses (and also Luke 9:38):
Luke 7:12a (ESV)
As he drew near to the gate of the town, behold, a man who had died was being carried out, the only son of his mother, and she was a widow, ...
Luke 8:42a (ESV)
for he had an only daughter, about twelve years of age, and she was dying.
... the context is quite clearly that of an only child, not an only child "that had a physical birth" rather than some other means. The word is also used by the same author earlier in John to refer, not to a child at all, but just to emphasis the "only" bit. In fact is used used of God himself:
John 1:18 (ESV)
No one has ever seen God; the only God, who is at the Father's side, he has made him known.
Another way to approach this issue is to show the other aspects of the relationship between Jesus and God. In John just a few chapters later we find Jesus making this statement:
John 10:30 (ESV)
I and the Father are one.
This (and a myriad of other verses indicating this unity of identity (see proof texts for Jesus being God)) lead us to a special understanding of who Jesus was in relation to God. We also find that this relationship between the three persons of the trinity is exclusive of other persons. When it comes to Lucifer, we find no similar references of a unity relationship with the father. In fact he is referenced as an angel, a created being.
1 ESV, HCSB, NIV, NLT, ISV, GW, BBE, WEB, and others...