In the Hebrew Bible, a marriage ceremony is never actually described.
True...hm...let me edit that...Song of Solomon might be one such description (if very poetic...but still might be or have a description of a wedding).
Priests and their attendants (Kohanim and Levites) are responsible for
work in the sanctuary, such as sacrifices and offerings, so although
they're the people "consecrated to God" and act as intermediaries
between the people and God, they don't have any part in helping a man
and woman form a permanent union as "one flesh" and become husband and
wife.
Hadn't thought of that, but it's true. Nothing in the law, that I remember.
In Genesis, a union between man and woman is described as just a
commitment that two people make to live together, help each other, "be
fruitful and multiply."
But society around them was involved in the process...
Isaac brought Rebecca "into his tent," and she became his wife, so in
their case, marriage is initiated between a couple when they engage in
sex.
That's not the only thing that happened. Remember that Abraham sent his servant to look for a wife for Isaac (made him swear he would go and find a wife, but not take his son there). Therefore, Abraham had to make something known to his servant, and this servant to the girl (while near a well, with lots of people around), and this girl and servant to her relatives, and THEN Rebecca accepted and WENT AAAAALL THE WAY to Abraham and Isaac's homestead to meet this groom.
So the thing wasn't: "Just have sex and become my wife." There was a very important process before sex "happened."
Besides, Isaac's mother having died, and the society around Abraham and Isacc knowing this, it must have been "prutty" obvious that something "was up" with Isaac taking this girl (which Abraham sent his servant to look for!) into what used to be his mother's tent (Gen. 24:67)!
Afterwards, they are understood as being bound to each other, and
thus living in a marriage. This is understandable, since biblical
times had a very different view of sex than what we see in modern
society.
Not too sure what you mean by "a different view of sex," but all of this makes a strong point for what is now called "traditional" wedding ceremonies: The acknowledgement of the couple by THEIR "society."
Since a marriage ceremony is never described in the Bible, what
qualities would make a concubine different than a wife?
Status? Recognition as such of the society around them? At least recognizion as such of those who were in charge of writing the events (Scribes? Priests?).
[...] is it just suggesting a social
status difference? A handmaid, even if she took up permanent residence
with one man, would never be a "wife", only a "concubine"?
"Bingo." Most likely.
Bilhah and Zilphah are described as Jacob's "concubines", and Rachel
and Leah are described as "wives", although the children from all four
women are equal since they all become the twelve tribes. So why not
call Rachel and Leah's handmaids wives?
Because they were servants of the wives (which Jacob worked 14 years to have), which the wives themselves gave to Jacob, eager to "have" children when one couldn't and the other one could...lots o' competition between some women...