None of the low-level graphics operations pay attention to the hold state; they just add to (or modify) what is there already. The hold state is only paid attention to by the high-level plotting operations such as plot() and surf(), which contain code that specifically detects the hold state and clears the axes if the hold is not active.
When you copyobj() you are not invoking any high-level graphics operations, so the object is just added to whatever is there already.
You should cla() as needed.
You should also re-write the code so that every graphics operation is specific about the object it is going to act on.
ax = subplot(2,1,1)
fimplicit(ax, @(x,y) (x-position(1)).^2+(y-position(2)).^2-r.^2,'r')
r = r + 0.1; position = position + [0.1,0];
axis(ax, 'equal'); axis(ax, [-10 10 -10 10]);
zoomed = subplot(2,1,2);
cla(zoomed);
copyobj(get(ax,'Children'),zoomed)
axis(zoomed, 'equal'); axis(zoomed, [-1 1 -1 1]);
pause(0.1)
counter = counter + 1;
