Do there exist any two ordered fields $F_1,F_2$ such that $F_1$ admits two distinct ordered field embeddings $f,g:F_1\rightrightarrows F_2$?
Note that $\mathbb{Q}[\sqrt 2]\rightrightarrows\mathbb{R}$ doesn't work, for example, because one of the embeddings doesn't respect the ordering.
Sure. For instance, let $F_1=F_2=\mathbb{Q}(x)$, ordered such that $x$ is infinitely large. Then there is the identity embedding $F_1\to F_2$, but there is also an embedding $F_1\to F_2$ that maps $x$ to $x+1$ (or, to any other infinitely large element).
On the other hand, there is no example where $F_2$ is Archimedean. Indeed, if $F_2$ is Archimedean then any field $F_1$ that embeds into it is also Archimedean, and then an embedding $F_1\to F_2$ is uniquely determined since it must fix the rationals and then every element is determined by the Dedekind cut it determines in the rationals.