10uV is inside the noise specs of really good linear PSUs. You cannot measure it with any reasonable certainty using cheap DMMs (ACV).
A Siglent SDG2042X (€475+VAT) will outperform any DSO's integrated AWG, and the specified amplitude accuracy is ±(1%+1mV), so 1V ± 0.02V (at 10KHz, and the uncertainty goes up with frequency). 1.00001V means 0.001% or 10ppm accuracy.
I don't think it's happening anytime soon. Although there might be audio analyzers with signal generators that reach those levels, a general purpose AWG will not. As a matter of fact, the resolution you asked for could not be resolved even with 16 bit ENOB, as that would be 65,536 steps, and around 15uV for a 1V peak to peak signal.