If one sets a default font for everything in .Xdefaults, or use
for example CDE, KDE (and perhaps Gnome) tools for setting style,
the tooltip will pick up that default font.
I meant a decent setup, not the default. I once proposed to make the
default font for the tooltips smaller than the one chosen for the
default face, but was told that there are problems to do that with X,
and I don't know enough about X to suggest a solution.
FWIW, my .emacs selects a much smaller font for tooltips.
The alternative is to have a long tooltip. Then one has to wait for
the tooltip to appear, shift focus of attention to the tooltip (which
may be on the other side of the screen), and then shift attention back
to the menu.
That's how tooltips work on all GUI platforms I've seen; Emacs just
tries to be compatible here. (Of course, you have variables to
customize the time delay for pop-up and pop-down, if the defaults
annoy you.)