Mistakes in Bredon's book "Topology and Geometry"?

Star (in older topology texts) often indicate torsion product of abelian groups, that is, $A * B := \operatorname{Tor}_{\Bbb Z}(A, B)$. Usually it is clear from the context whether free product or torsion product is meant.

I think that you are missing the definition of 'orientable along $A$'. I haven't got that book of Bredon to hand, but presumably 'orientable along $A$' means that if you move a local orientation of $M$ around a closed path that stays in $A$ then it will come back to the same local orientation. In particular, in the case when $A$ is a single point, then $M$ will always be orientable along $A$, regardless of whether $M$ is orientable or not, so the case that you view as wrong doesn't arise.

I agree with Denis T's interpretation of the notation $A*B$.