There is a temptation into which some of us fall (not necessarily you, OP); namely, to think because we live in the 21st century and have appended ostensibly "modern" labels (such as cognitive dissonance, bystander effect, déjà vu, buyer's remorse, Freudian Slip--well, you get the idea) to human behavior, that that behavior did not exist in previous centuries. I guess you'd call it a form of anachronistic thinking. (Would it be the opposite of anachronism?)
Do we really think, for example, that a teenage girl in 1423 never said to her father who just criticized her for going out in public with her bare ankles showing,
"Oh, daddy, you're soooooo old fashioned. For goodness' sake, this is the 15th century! Get with it!"
The point is: the labels and vocabulary for describing behavior may change; the behavior, "not so much" (as the young people say nowadays).
In answer to one of your questions, then: no, Paul most likely was not consciously singling out one particular person to intervene in an unpleasant situation into which no one but he, apparently, was willing to insert himself, in order to combat the "bystander effect." Paul was simply enlisting the help of a
true yokefellow, helper, companion, teammate
faithful partner, friend, yokefellow (or yoke-fellow)
to intervene in the sticky situation created by Euodia and Syntyche. He was encouraging a "good Samaritan" to step forward and minister to two sisters in the Lord who were evidently butting heads, thinking unanimity is somehow better than unity!
Now, was Paul singling out an unnamed "true companion" (NASB), or was he singling out a brother in the Lord by the name of Syzygus? Allow me to digress.
The brothers Gershwin wrote a song for the 1937 film "Shall We Dance," and Fred Astair and Ginger Rogers sang the song while roller-skating in the film. The song was called "Let's Call the Whole Thing Off."
A famous line from the song goes,
“You like to-may-toes /təˈmeɪtoʊz/ and I like to-mah-toes /təˈmɑːtoʊz/”
The song goes on to parody the different ways--depending on one's regional dialect--of saying various words (e.g., eether/eyether, neether/nyther; potato/potahto; pajamas/pajahmas), and since Ginger and Fred can't seem to agree on how to pronounce the words in question, then maybe they should simply break off their romantic relationship!
In Philippians 4:2, we have a somewhat similar situation, but with a twist. Paul, in attempting to resolve that interpersonal conflict between two sisters in the Lord, whom he singles out by name, engages the help of either
- a true companion (from Gk σύζυγος/suzugos)
- a brother in the Lord named Syzygos (with a capital Sigma as the initial letter).
Instead of calling the whole thing off (because some scholars say suzugos and some scholars say Syzygos), perhaps the sensible thing to do is to say simply, "It makes not a whit of difference, at least in the grand scheme of things!"
To me, in light of Paul's singling out Euodia and Syntyche by name, it seems reasonable to assume Paul was addressing perhaps the lead elder of the church at Philippi, or at least a brother in the Lord with whom Paul was familiar and with whom he had perhaps worked side by side in the past.
If, on the other hand, Paul was simply addressing a true companion whom he did not have to name (the ol' "you know who you are"), that exegesis is also perfectly reasonable.
Either way, no key doctrine of Scripture is in danger if the word in question is suzugos or Syzygos. In other words, let's not call the whole thing off! Just put in your nickel and make your choice!