Wire gauge? AWG 34H? meaning?

Magnet wire is often called "varnished" or "enameled," but it is actually insulated with a thin coating of insulating material designed specifically for the purpose. I assume that "heavy poly nylon" material is one of those materials.

The "H" is probably the temperature class of the insulation. Class H insulation is rated to operate at up to 180 degrees C.

Edit 1

The "H" could also stand for "heavy" as in the "heavy build" or "double build" terminology used to describe thicker than standard insulation. It is part of a specific manufacturer's catalog number rather than a part of the American Wire Gauge (AWG) designation.

Manufacturer's sell many different varieties of magnet wire. The manufacturer's literature and NEMA or IEC standards provide details regarding the insulation materials used and the electrical and mechanical characteristics of the wire.