For every $t\geqslant0$, the distribution of $(X_{t+s})_{s\geqslant0}$ conditionally on $\sigma(X_u;u\leqslant t)$ is the Dirac distribution at $(X_t+s)_{s\geqslant0}$ on $[X_t\ne0]$ and is $\mu$ on $[X_t=0]$, where $\mu$ denotes the (unconditional) distribution of $(X_s)_{s\geqslant0}$. Thus the distribution of $(X_{t+s})_{s\geqslant0}$ conditionally on $\sigma(X_u;u\leqslant t)$ depends on $X_t$ only and $(X_t)_{t\geqslant0}$ is a Markov process.
On the other hand, $\tau$ is finite almost surely and the distribution of $(X_{\tau+s})_{s\geqslant0}$ conditionally on the past of $\tau$ is the Dirac distribution at $\xi$, where $\xi:s\mapsto s$. This is not $\mu$ hence $(X_t)_{t\geqslant0}$ is not a strong Markov process.