Say this happens at the point $(a,b)$. Then the slope of the tangent line that point is also $b$, so the tangent line at that point is
which can be rewritten as
That is, the tangent line at $(a,b)$ is the line of slope $b$ which intersects the $x$-axis at the point $(a-1, 0)$.
So, if you linearly approximate $f$ based on its behavior at $(a,b)$, your approximation will vanish one $x$-unit before $a$. In some cases, this might mean your chosen units for $x$ and $f(x)$ are in some way compatible near $(a,b)$. However, it can't mean anything more physical than that, because if you change units then this condition will no longer be true!
A similar question you could ask that might be a little more meaningful (because it does continue to be true after a change of units) is:
What is the significance of a point $x_0$ where $f(x_0)=x_0f'(x_0)$?
By the same kind of computation I did above, you can check that this means the tangent line to $f$ at $x_0$ passes through the origin.