Can a stochastic process correspond to multiple SDE's?

Question

In this question, and this question, it is clear that the solution of the one-dimensional SDE $dX_t = \frac{1}{2 X_t} dt + dB_t$ is $X_t-X_0=B_t$. This is also the Bessel process. I am confused by the solution, as it is equivalent ti $d X_t = dB_t$. How can there be two equivalent SDE representations of the same stochastic process $X_t=W_t$?