Machatonim is a word that, so far as I know, has no equivalent in any language except Hebrew/Yiddish. The parents of the people your children marry are your machatonim (the male is your mechuten, and the female your machatonister).
They have an official status. Theyre not just the couple who happen to be the parents of the boy/girl your child happened to marry, whom you will only see at the grandchildrens britot and weddings, if then.
They are people with whom you enter into a lifelong relationship that is dignified by a title by virtue of your respective childrens union. Surely, theres something very Jewish about that.
Machatonim is derived from chatan which means bridegroom or son-in-law. Chatunah is the Hebrew word for wedding. Chatan also has the connotation of guest of honour, or star of the day as in chatan Torah and chatan Bereishit, the people honoured to be called up at the end and the beginning of the Torah on Simchat Torah.
The verb chatan, means to tie, connect or covenant. (Jastrows Dictionary says that it is related to the Assyrian word chatanu meaning to protect).
We can see from this etymology the sense of a marriage being, not just a holy covenant between two people, but also an alliance between two families, a meaning that is preserved in the endurance of the word machatonim until today.