Almost all paths $t\mapsto B_t(\omega)$ of the Wiener process are in fact not differentiable. Therefore, if we take $B_t$ as position of the particle $x(t)$ we end up having an unphysical particle whose velocity is not well defined. As far as I know, this led  Langevin, Ornstein and Uhlenbeck to develop a dynamical theory of Brownian motion by which the velocity $v(t)=\dot x(t)$ is 'driven' by a Wiener process
$$
v(t)=e^{-\beta t}v_0+e^{-\beta t}\int_0^te^{\beta s}\,dB_s
$$
which is described in Nelson [1]. The parameter $\beta$ describes friction. The larger that is the faster the velocity tends to zero. If $\beta=0$ we have
$$
v(t)=v_0+B_t\,,\quad\quad x(t)=x_0+\int_0^tv(s)\,ds=x_0+v_0t+\int_0^tB_s\,ds\,.
$$
In this theory the mean (over many particles) of $x(t)v(t)$ is
\begin{align}
\mathbb E[x(t)v(t)]&=\mathbb E\left[B_t\int_0^tB_s\,ds\right]=\mathbb E\left[B_t\int_0^t(t-s)\,dB_s\right]=\int_0^t(t-s)\,ds=t^2/2.
\end{align}
Note that the Langevin/OU theory is not compatible with $\langle x^2\rangle=2Dt$.
That equation is due to Einstein and Smoluchovski which Nelson calls a
highly idealized treatment in his Chapter 9. In fact in Nelson's notation, Einstein Smoluchovski corresponds to
$$
x(t)=\sqrt{2D}B_t
$$
which Langevin/OU discarded as unphysical. Nelson also writes that the predictions of the OU theory are numerically indistinguishable of the Einstein Smoluchovsik theory.

[1] E. Nelson, [*Dynamical Theory of Brownian Motion*](https://web.math.princeton.edu/~nelson/books/bmotion.pdf)