This answer is simply to write the details for my comment above. It amounts to doing a little more work with homotopy equivalences, so as to carry out essentially the argument you gave in your comment regarding the smooth case.
Assume that $f:X\to Y$ is a weak equivalence of topological spaces. We want to show that the map induced on topological bordism (as you define it) is a bijection.
Regarding the surjectivity, I will simply repeat the content of Tom Goodwillie's comment. First, we observe that any topological manifold has the homotopy type of a CW-complex (you can find references to proofs by Hanner in Milnor's "On spaces having the homotopy type of a CW-complex", as remarked in the comment of Greg Friedman). It follows that $f$ induces a bijection on homotopy classes of maps $f_\ast :[M,X]\to [M,Y]$. Therefore, any map $g:M\to Y$ is homotopic and thus cobordant to a map which lifts to $X$. This proves surjectivity.
To prove injectivity, we need to know that the inclusion of the boundary of a topological manifold is a closed Hurewicz cofibration, that is a NDR-pair. This holds since the boundary of a topological manifold is collared (see chapter 2 of Ferry's notes). With this fact at hand, we can show that for any topological manifold $M$, the pair $(M,\partial M)$ is homotopy equivalent rel $\partial M$ to a pair $(A,\partial M)$ such that the inclusion $\partial M\to A$ is a Serre cofibration.
Proof of claim:
Simply factor the inclusion $\partial M\to M$ as a Serre cofibration followed by a weak equivalence $\partial M\to A\to M$. Since $\partial M$ is a topological manifold, it has the homotopy type of a CW-complex, and it follows that $A$ also has the homotopy type of a CW-complex. Then the map $A\to M$ is a weak equivalence of spaces with the homotopy type of CW-complexes, and thus a homotopy equivalence. The map $A\to M$ is therefore a homotopy equivalence rel $\partial M$, since both maps $\partial M\to A$ and $\partial M \to M$ are Hurewicz cofibrations.
We can now prove injectivity by carrying out an argument similar to the one you wrote in your comment. Given bordism classes $g_0:M_0\to X$, $g_1:M_1\to X$ in $X$ and a cobordism $g:M\to Y$ ($\partial M=M_0\coprod M_1$) between $f\circ g_0$ and $f\circ g_1$, we need to find a cobordism $g':M\to X$ between $g_0$ and $g_1$. For that purpose, we replace $(M,\partial M)$ with $(A,\partial M)$ as described above. Since $\partial M\to A$ is a Serre cofibration, we can find a lift up to homotopy $A\to X$ of the composite $A\to M\to Y$, which furthermore extends the map $(g_0,g_1):\partial M\to X$. Composing with the homotopy equivalence $M\to A$ rel $\partial M$, we get the desired map $g':M\to X$.
Best Answer
Yes, there is a very nice geometric interpretation over any manifold $X$ satisfying $\chi(X)=0$, for a version of Reidemeister torsion spelled out in Turaev's paper "Euler structures, nonsingular vector fields, and torsions of Reidemeister type". Ian Agol's comment mentioned the Seiberg-Witten invariants, which equals this version of torsion in 3 dimensions. It turns out that both of these are equal to an $S^1$-valued Morse theoretic invariant $I(X)$ defined by Hutchings and Lee in their PhD work:
"Circle-valued Morse theory and Reidemeister torsion" (Geom. Top. Vol.3, 1999)
Roughly speaking, we equip $X$ with a suitable Morse function $f:X\to S^1$ for which all critical points and closed orbits of $-\nabla f$ are nondegenerate. Then $I(X)$ is defined by suitably counting the Morse-flowlines between critical points as well as the closed periodic orbits (remembering their periods). The invariant is independent of the choice of $f$ and it is identified with the Redeimester torsion (for a suitable homology orientation). When $\dim X=3$ and $b_1(X)>0$, this also recovers the Seiberg-Witten invariants. We can view $I(X)$ as the analog of the Gromov invariant in 4 dimensions which suitably counts $J$-holomorphic curves.
Here is a simple example which may make the "involvement of products of determinants of basis-change matrices" less opaque to you. When $f:X\to S^1$ is a fiber bundle with fiber $\Sigma$, the periodic flow of $-\nabla f$ defines a self-map $\varphi:\Sigma\to\Sigma$, and $I(X)$ effectively counts fixed points of $\varphi^k$ weighted by their Lefschetz-sign. The identification with torsion translates into the Lefschetz fixed-point theorem.