I attempted to create an example showing the behavior of set_units when shared axes have units, but there are many problems.

1. The units examples rely on TaggedValue defined in basic_units.py. TaggedValue is a container based on numpy's dispatch mechanism where __array__ returns the underlying data as a base class ndarray - no more units. This breaks plotting functions like plot that, very early in execution, calls cbook._check_1d that calls np.atleast1d that calls __array__. Subsequent call to set_units then fails because the data, now ndarray has no units converter.
2. Plotting functions like bar and scatter don't assign a 'units' callback, so the call to Axis.set_units does nothing. Unlike plot that does set a 'units' callback.
3. I can change TaggedValue.__array__ to return an array of TaggedValue objects, but this breaks the ellipse example.

@jklymak Is this proposed PR sufficient? Creating an example using the existing basic_units definition isn't going to work as discussed above. I read the units discourse and see that units has a rather long history. Is there any work on defining an interface layer between incoming data and plot function internals?