What the book you are quoting is almost true. It has in fact become very
popular in applied mathematics to identify solutions to the Langevin SDE
with Gauss-Markov processes.
The exact story goes as follows
According to an 80 year old theorem by J.L. Doob [1], [2] every stationary Gauss-Markov process $Y_t$ with continuous sample paths has covariance function
$$\tag{1}
{\rm Cov}\big[Y_t,Y_s\big]=\frac{\sigma^2}{2\beta}e^{-\beta|t-s|}\,.
$$
Doob writes $\sigma_0^2=\frac{\sigma^2}{2\beta}$ and calls this $Y_t$ an O.U. process.
Ornstein and Uhlenbeck themselves have instead studied the meanreverting Langevin type SDE
$$\tag{2}
dX_t=\beta\,(\mu-X_t)\,dt+\sigma\,dW_t
$$
which gained a lot of popularity for example in quantitative finance.
Its explicit solution is
$$
X_t=X_0\,e^{-\beta t}+\mu(1-e^{-\beta t})+e^{-\beta t}\int_0^te^{\beta s}\,dW_s\,.
$$
When $X_0=0$ and $\mu=0$ there is a close relationship between $X$ and $Y$:
Namely, Doob shows in [1] also that
there exists a standard Brownian motion $B_t$ such that
$$
B_{\sigma^2_0t}=\sqrt{t}\,Y_{\textstyle\frac{\log t}{2\beta}}\,
$$
holds.
Changing the time variable to $s=\frac{\log t}{2\beta}$ this can be written as
$$\tag{3}
Y_s=e^{-\beta s}B_{T_s}\,,\quad\text{ where }\quad T_s:=\textstyle\sigma^2_0\, e^{2\beta\, s}\,.
$$
Lemma. The process $Y_t$ has constant variance $\sigma^2_0$ while
$$\tag{4}
X_t:=e^{-\beta t}B_{(T_t\color{red}{-\,\sigma^2_0})}
$$
solves the special case of the SDE (2) with $X_0=0$ and $\mu=0\,.$
Proof.
The time changed Brownian motion in (4) can be written as
$$\tag{5}
B_{(T_t\color{red}{-\,\sigma^2_0})}=\color{red}{\sigma}\int_0^te^{\beta s}\,dW_s
$$
with yet another standard Brownian motion $W$. To see this note that
$$
T_t-\,\sigma_0^2=\sigma^2_0\, e^{2\beta\, t}-\sigma_0^2=\sigma^2\frac{e^{2\beta t}-1}{2\beta}=\sigma^2\int_0^te^{2\beta s}\,ds\,.
$$
Therefore,
\begin{align}
W_t=\int_0^t\frac{e^{-\beta\,u}}{\sigma}\,dB_{(T_u-\,\sigma^2_0)}=
\int_0^t\frac{e^{-\beta\,u}}{\sigma}\,dB_{\textstyle(\sigma^2\int_0^ue^{2\beta s}\,ds)}
\end{align}
which is seen to be a continuous martingale with quadratic variation $t$ and therefore
a Brownian motion.
Combining (4) and (5) we get
$$
X_t=\sigma\,e^{-\beta t}\int_0^te^{\beta s}\,dW_s
$$
which is seen to solve the familiar SDE
$$
dX_t=-\beta X_t\,dt+\sigma\,dW_t\,.
$$
Few remarks:
It is easy to see that $X_t$ is Gauss-Markov but not stationary.
Its covariance function is
$$\tag{6}
{\rm Cov}\big[X_t,X_s\big]=\sigma^2e^{-\beta(t+s)}\frac{e^{-2\beta(t\wedge s)}-1}{2\beta}=\sigma^2\frac{e^{-\beta|t-s|}-e^{-\beta(t+s)}}{2\beta}
$$
(cf. [2]).
The Brownian bridge
$$
Z_t=W_t-\frac{tW_T}{T}\,,\quad 0\le t\le T\,,
$$
is continuous, Gaussian and Markov with respect to its own filtration [3]
and has covariance function
$$\tag{7}
{\rm Cov}\big[Z_t,Z_s\big]=t\wedge s-\frac{ts}{T}\,.
$$
But it is not stationary and therefore not an O.U. process $Y$ in the sense of Doob [1]. The Brownian bridge does also not satisfy the Langevin SDE (2) because its covariance function cannot be achieved by (6). (For $\beta\to 0$
the expression in (6) converges to $\sigma^2(t\wedge s)$, not to (7).)
[1] Doob, J.L. (April 1942). The Brownian Movement and Stochastic Equations. Annals of Mathematics. JSTOR. 43 (2): 351–369. doi:10.2307/1968873. ISSN 0003-486X. JSTOR 1968873.
[2] Wikipedia Ornstein Uhlenbeck Process.
[3] Borisov, I.S. On a Criterion for Gaussian Random Processes to Be Markovian. https://doi.org/10.1137/1127097
