Is there a name for this category associated to any functor between categories?

It sounds like you're looking for the collage category of the profunctor $F_*:C\not\to D,\ (c,d)\mapsto \hom_D(Fc,d)$ induced by the functor $F$:

There we freely add arrows $c\to Fc$ subject to making the following squares commutative for each arrow $u:c\to c'$: $$\matrix{c &\overset u\to& c'\\ \downarrow &&\downarrow \\ Fc& \underset{Fu}\to &Fc'}$$

Consider the category of categories $E$ equipped with functors: \begin{eqnarray*}G_E\colon C\to E,\\H_E\colon D\to E,\end{eqnarray*} and a natural transformation $\eta_E\colon G_E\to H_E\circ F$.

Let morphisms in this category be functors $K\colon E_1\to E_2$ satisfying:
\begin{eqnarray*}KG_{E_1}&=&G_{E_2}&,\\KH_{E_1}&=&H_{E_2}&,\\K(\eta_{E_1})&=&\eta_{E_2}.\end{eqnarray*}

Then your category along with natural inclusions $C\hookrightarrow E,\,D\hookrightarrow E$ is the initial object in this category.

This category is also known as a co-comma, which is a type of pushout of the span $D\leftarrow C\to C$. Concretely, it given by taking the weighted colimit with weight $\bullet \to (\bullet \to \bullet)\leftarrow \bullet$, where the first inclusion is at the terminal and the second at the initial object. This gives your construction’s universal property: it represents triples $(f,g,\alpha)$ of a functor $f:C\to E,$ a functor $g:D\to E$, and a natural transformation$ \alpha:f \Rightarrow g\circ F$. It is not precisely a lax pushout, although they’re closely related.

I don’t know of a great literature reference for co-commas among categories, but there are a couple of relevant questions here and on MO:

https://mathoverflow.net/questions/247280/an-explicit-description-of-cocomma-categories

How Co-Comma Categories are constructed?