I think the term fifth wheel is a term for something superfluous, akin to an extra wheel on a four-wheeled carriage. The Wikipedia claim about the fifth wheel referring to the the fifth-wheel coupling on trailers seems far fetched, since it may be fifth, but it's by no means useless.
I offer as evidence a play called The Fifth Wheel, A Comedy in Three Acts, published in 1869 and owned by the De Witt Publishing House. This appears to be a comedy of manners about love and wealth, and its final lines (addressed to the audience) are as follows:
Vande. It’s a blessing you’re all so happy that you forget Joralemon
and me. Poor fifth wheels! utterly useless1 — but we still form
part of the family coach2.
Joral. By pairs1 both wheels and human beings move,
Mrs. B. The dual nature of our lives to prove,
Cook. We share the downhill,
Miss Sp: Uphill roads of life,
Vande. Joined by the axle Love, true man and wife!
Wheel takes the dust, the mire, the sun3 —
Brain. But for the lesson’s sake, grant it its run4!
- Useful wheels are paired, but the unpaired fifth wheel is "utterly
useless", unlike a fifth-wheel coupling
- Fifth-wheel couplings are for vehicles that haul freight, not for a "family coach".
- Fifth-wheel couplings don't take dirt, mud, and sun; they're covered by the trailer.
- Fifth-wheel couplings cannot be granted a "run"; they're fixed couplings and don't turn.