From Simon Withers (10.05):

One last word on this problem from a friend who I asked about it. I think
he's put his finger on the nub of the question quite well:
"the real question is "Why anyone would want to design a block
set with only one solution that prevents mistakes." The discovery
process usually begins by getting a block in the wrong hole or using
a hammer and getting one jammed."

From Simon Withers (09.05):

You ask: "You know those wooden boxes for putting wooden shapes in
(cylinder, oblong box, star, etc), well, my son's has eight holes, and none
of the wooden shapes fit through the wrong hole. Mathematically, what is the
maximum number of holes possible? (Also the maxiumum given the shapes have a maximum of twelve edges, which the cross has.) This number surely can't be
infinite, but how can it be calculated? (Aug 05)"

Well, it is infinite IF you are allowed to have the same shape multiple
times. Think of your oblong. Start with one oblong which is only slightly
wider than it is tall. Now add a second oblong which is slightly wider and
slightly less tall. Now add a third in the same way and repeat an infinite
number of times. The template for your toy now looks like an infinitely
large version figure 1 in the attached diagram.

None of these shapes will fit into any of the other holes so it meets all
your criteria.

However, if you insist on having a "different" shape for each hole then it's
slightly more complex... What exactly consitutes a "different shape"?
If a different shape must have a different number of edges, then the answer
must be ten. It clearly can't be more than twelve, because we would run out
of "new" shapes at that point. And we can't cut a shape using just one or
two edges, however fine our chisel, so the maximum possible is ten (unless
we are allowed curved edges in which case we can add in the circle and the
oval?). We can then do a similar thing to the case above to get our ten
holes, none of which will accommodate the same shape twice. Start off with a
regular, equilateral triangle, then do a slightly wider but slightly less
tall oblong, then a wider still and less tall pentangle etc. I tried to draw
this but my drawing skills let me down badly!

If two shapes are allowed to share the same number of edges but must have
different internal angles, then we're back to infinity again. Going back to
our wider and less tall oblongs, we can just extend two of the sides on each
oblong to make one corner increasingly "pointy". Each object now has a
different set of internal angles, but we can still handle an infinite number
of them. See figure 2.
The problem here is that these shapes are a bit too weird for a child's toy,
and I suspect yours are in some way "regular". If this is the case then we
need to define "regularity". One definition would be to insist on each side
of a shape being of the same length. Unfortunately that would preclude your
oblong, so your own toy would not comply with this rule. A square would be
your only allowed four-sided shape. And even if this were our rule, we could
use increasingly wide and less tall rhomboids, as per figure 3.
My rhomboids are still a bit too weird for our child's toy though, so I
think we need another definition of regularity. If you can come up with your
own definition we can have another crack at this, but for now, I think your
answer must be infinity.

Interesting thought experiment, on which I've now spend way, way longer than
I'd intended.