They both relate in one way or the other to extensions of the same mathematical object, namely the Cauchy-Euler differential equation, that has the form
$$a_{n} x^n f^{(n)}(x) + a_{n-1} x^{n-1} f^{(n-1)}(x) + \cdots + a_0 f(x) = 0 \tag{1}$$
where $f^{(n)}$ denotes the $n$-th derivative of the function $f$. Its 1st-order version is
$$a_{1} x f^{(1)}(x) + a_0 f(x) = 0\tag{2}$$
A. "Eulers theorem for homogeneous functions".
Consider the 1st-order Cauchy-Euler equation, in a multivariate extension:
$$ a_1\mathbf x'\cdot \nabla f(\mathbf x) + a_0f(\mathbf x) = 0 \tag{3}$$
Euler's theorem for homogeneous functions says essentially that if a multivariate function is homogeneous of degree $r$, then it satisfies the multivariate first-order Cauchy-Euler equation, with $a_1 = -1, a_0 =r$.
B. "Euler's equation in consumption."
Consider a univariate but non-linear extension for the 1st-order Cauchy-Euler differential equation:
$$a_{1}g(x) f^{(1)}(x) + a_0 f(x) = 0 \tag{4}$$
Now set $x=t$ (i.e. equal to time), and $f(x) = C(t)$ (say, per capita consumption). Then $(4)$ is written
$$a_{1}g(t) \frac{dC(t)}{dt} + a_0 C(t) = 0 $$
$$\implies \dot C = -\frac {a_0}{a_{1}g(t)}C(t) \tag{5}$$
What we realize is that the solution to the intertemporal utility maximization problem is described by an equation of the form of $(5)$, with suitable values and forms for $a_0, a_1, g(t)$. For example, in the Ramsey model, we could set $a_0=-1, a_1 = 1, g(t) = (r(t)-\rho)^{-1}$
The discrete analogue (i.e. for discrete time) of the Cauchy-Euler non-linear extension is
$$a_{1}g(t) \Delta C_{t+1} + a_0 C_t = 0 $$
$$\implies C_{t+1} = \left (1-\frac {a_0}{a_1 g(t)}\right) C_t$$
and again we see that the solution to the intertemporal utility problem in discrete time satisfies a Cauchy-Euler 1st order equation.
Why the "Cauchy" name prefix has dropped out in the economics literature, I don't know.
