The solution in the book doesn't consider the recoil velocity $u$ as relativistic. So, no need of $\gamma$. But they consider simply that the mass of the atom increases, instead of $m_0$ to $m$. So, this is what they do, I mean, their equations of conservation:
(1) $h \nu /c = mu, \ \ \ $ linear momentum conservation,
(2) $(m_0c^2 + h\nu)^2 = m^2c^4 + m^2u^2c^2, \ \ \ $ energy conservation and Klein-Gordon equation.
Now, from eq. (1) I obtain
$u = \frac {h \nu}{mc}$,
and I introduce in eq. (2).
$(m_0c^2 + h\nu)^2 = m^2c^4 + h^2 \nu ^2$.
From the last equation we can extract $m$,
$m_0^2 + \frac {2h\nu m_0}{c^2} = m^2$.
The quantity on the LHS can be eventually completed to a square
$m_0^2 + \frac {2h\nu m_0}{c^2} + \frac {(h \nu)^2}{c^4} = m^2$.
One way I see is to take the limit of infinitesimally small $\Delta d$, $\Delta t$, and $\Delta E_k$. Then, resuming from this step:
$\Delta E_k \frac{\Delta t}{\Delta d} = mv$
taking the limit of small $t$, $d$, and $\Delta E_k$, re-write them as differential units, and use $ dd / d t = d v$ (forgive the awkwardness of the differential unit of distance being $dd$):
$ \frac{ d E_k}{d v} = mv $
And then,
$ d E_k = mv \ d v$
Integrate both sides,
$\int d E_k = \int m v d v = \frac{mv^2}{2}= E_k$
As we had hoped to show!
Best Answer
In short, no, it is not a coincidence, they are related. Namely, you may derive the kinetic energy as the first order approximation to the relativistic energy.
We have,
$$ E_0 = mc^2 $$
as you say correctly. Then
$$ E = \gamma m c^2 = \left( 1 - \frac{v^2}{c^2}\right)^{-\frac{1}{2}} m c^2 $$
or using a binomial expansion
$$ E \simeq \left( 1 + \frac{1}{2} \frac{v^2}{c^2} + \dots \right) m c^2 \simeq mc^2 + \frac{1}{2} mv^2 $$
So subtacting the rest energy $E_0$ we get
$$ E_k = E - E_0 \simeq \left( mc^2 + \frac{1}{2} mv^2 \right) - \left( mc^2 \right) = \frac{1}{2} m v^2$$
Note that we can of course only use this expansion when $v \ll c$. This makes sense, because that is exactly the case in Newtonian mechanics, which is where we use the more familiar kinetic energy formula.