Very small transformer powering a microwave oven

If you double-check your wiring diagram you will find that is NOT a transformer. That is a common mode choke to keep RFI from being transmitted out through the mains power cord.

There is no way (with currently-known technology) to make a mains-frequency transformer that small that handles as much current as a microwave oven draws. You have seen how big the transformer needs to be by your comparison with the size of the high-voltage transformer.

Ref: https://en.wikipedia.org/wiki/Choke_(electronics)#Common-mode_chokes

As an amplification of Richard Crowley's answer, I'd do this in a comment but there is no way to include a circuit diagram. Your circuit actually looks like

^{simulate this circuit – Schematic created using CircuitLab}

as far as the "transformer" is concerned. As stated, it's a common-mode filter which keeps the microwave from driving noise onto the power lines.

Most likely, R1 is a PTC surge preventer, although this is not guaranteed.
R1 (the resistor clearly visible in the OP photo with a green multiplier band) will be across C3 and is likely intended to bleed any residual voltage from it after AC supply is removed.