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Here is a simple-minded example that is probably not the sort of thing you are looking for since it is not defined in terms of maps. For a topological space $X$ define its "local homology spectrum" to be $$ \{ n\ |\ H_n(X,X-\{x\};{\mathbb Z})\neq 0 \hbox{ for some $x\in X$}\} $$ Then set $X\sim Y$ if $X$ and $Y$ have the same local homology spectrum. This is an equivalence relation under which all manifolds of the same dimension are equivalent, though they need not be homotopy equivalent. In the other direction, euclidean spaces of different dimensions are homotopy equivalent but are not equivalent in this new sense.