A wire can be embedded in 2 or 3 (or N) dimensions, and if the only degree of freedom for travelling along the wire is forward and back (no thickness), then it can also be considered a one-dimensional line.

I think the use of the word 'essentially' signifies that the definition of thickness depends on the domain you're examining. In this case, I'd expect we're talking about the connectedness of chemical bonds. No thickness should mean it's a chain of atoms each with only 2 bonds, to the atoms before and after it in the chain.

No expert here either, but I suppose that meant the wire has essentially only "length" (e.g. x dimension), and no "thickness" (y dimension)?

Usually, a wire is like a long thin filament. Surely, this is closer to one-dimensional than two-.