Ivan's comment is spot on. It take two to tango (or hydrogen bond) - a donor $\ce{Z-H}$, where $\ce{Z}$ is some electronegative atom (usually $\ce{O},\ \ce{N},$ or $\ce{F}$) and an acceptor (something with just $\ce{Z}$). 1-Butanol can hydrogen bond with itself, since it has an $\ce{O-H}$ group capable of being both donor and acceptor. Deithyl ether cannot hydrogen bond with itself since it lacks an $\ce{O-H}$ group. However, both can form hydrogen bonds with water. Thus, 1-butanol has the higher boiling point, but they both are similarly soluble in water.
Actually, Morrison and Boyd oversimplify the solubility difference (and there is one). 1-Butanol is slightly more soluble in water than diethyl ether, perhaps because it can interact with water in more than one way, although the dipole moment of the alcohol group is certainky larger than that of the ether group as well.
The NIH PubChem entries for diethyl ether and 1-butanol list several solubilities in water, but I want to focus on two entries that share the same source for both compounds:
Solubilities in water at $25\ ^\circ \mathrm{C}$ from the NIH TOXNET Hazardous Substances Database
$$\begin{array}{c|c|c|c}
\mathrm{substance} & \mathrm{Solubility\ in\ water\ (g/100\ mL)} & \mathrm{density\ (g/mL)} &\mathrm{volume\ percent}\\ \hline
\text{1-butanol} & 6.8 & 0.8098 & 8.4\\
\text{diethyl ether} & 6.0 & 0.7134 & 8.4
\end{array}$$
Solubilities in water at $20\ ^\circ \mathrm{C}$ from the International Labor Organization's International Chemical Safety Cards
$$\begin{array}{c|c|c|c}
\mathrm{substance} & \mathrm{Solubility\ in\ water\ (g/100\ mL)} & \mathrm{density\ (g/mL)} &\mathrm{volume\ percent}\\ \hline
\text{1-butanol} & 7.7 & 0.8098 & 9.5\\
\text{diethyl ether} & 6.9 & 0.7134 & 9.7
\end{array}$$
Note that I have used the solubility data to calculate percent compositions by volume in both cases. It was and still is common to report low solubilities as percents composition. It also is and was common to see percents composition listed without specifying whether they were mass/mass (which should be the default), mass/volume (convenient for dilute solutions), or volume/volume (convenient for solutions, but unhelpful due to colligative properties).
At both temperatures, the data supports that 1-butanol and diethyl ether have nearly the same percent solubility by volume. Morrison and Boyd is an older generation of textbook, first published before the world wide web. If they found a source of solubility data for these compounds that listed percents (but did not specify volume/volume), then the authors made a reasonable assumption that those percents were mass/mass or mass/volume and reported the same solubility.
The slight solubility edge for the n-alkanols persists over a larger range of compounds. The table below lists the solubilities in grams per 100 mL of water of the n-alkanol and the symmetric ether, with data drawn from the NIH Hazardous Substance Database.
$$\begin{array}{c|c|c|c|c}
\text{sol. in water in g/100 mL} & \ce{C2} & \ce{C4} & \ce{C6} & \ce{C8}\\ \hline
\text{n-alkanol} & \text{miscible} & 6.8 & 5.9 & 0.54 \\
\text{symmetric ether} & 4.6 & 6.0 & 4.9 & 0.30
\end{array}$$
