Copies of topological fundamental groups inside etale fundamental groups given by different embeddings of your field into $\mathbb{C}$

This is only a partial answer to your question, but I think I have an example where the images are not conjugate.

Assume that $K$ has a non-real embedding $\sigma : K \hookrightarrow \mathbb{C}$, and let $\iota$ denote complex conjugation, so that $\iota \circ \sigma$ is another non-real embedding. Let $X$ be the projective line over $K$ with the points $z_1, z_2, z_3, z_4 \in K$ deleted, where $\sigma(z_i) = \iota \circ \sigma(z_i) \in \mathbb{R}$. Assume that the $z_i$'s are ordered so that their images under $\sigma$ go from least to greatest. Now let $\gamma \in \pi_1^{\mathrm{top}}(X_{\mathbb{C}}(\mathbb{C}))$ be represented by a loop wrapping around the missing points $z_1$ and $z_3$ and passing below $z_2$. Then I believe it's fairly straightforward to show that

(i) its image under the automorphism $\iota_* : \pi_1^{\mathrm{top}}(X_{\mathbb{C}}(\mathbb{C})) \to \pi_1^{\mathrm{top}}(X_{\mathbb{C}}(\mathbb{C}))$ induced by $\iota$ is represented literally by the image of that loop reflected across the real axis (the reflection will wrap around $z_1$ and $z_3$ and pass above $z_2$); and

(ii) the image of $\gamma$ under the map $\sigma^* : \pi_1^{\mathrm{top}}(X_{\mathbb{C}}(\mathbb{C})) \to \pi_1^{\mathrm{\acute{e}t}}(X_{\bar{K}})$ that you described (induced by the embedding $\sigma$) is equal to the image of $\iota_*(\gamma)$ under the analogous map $(\iota \circ \sigma)^*$.

But the group elements $\gamma$ and $\iota_*(\gamma)$ are not equivalent up to conjugation; in fact, the elements they represent in $H_1(X_{\mathbb{C}}(\mathbb{C}), \mathbb{Z})$ differ by a sign. Or we can lift the loops up to the double cover which gives the complex elliptic curve ramified over $z_1, ... , z_4$ and see that they represent two distinct elements (no longer differing by a sign) of the homology group of that genus-1 torus.

If we want an example for a hyperbolic Riemann surface, we can do the same trick but with points $z_1, ... , z_{2g + 2}$ for some integer $g \geq 2$, so that we are comparing two loops representing distinct elements of the homology group of the genus-$g$ complex hyperelliptic curve ramified over the $z_i$'s.

EDIT: Hmm just realized that I didn't really answer any of your question, since you were asking about images of an entire group rather than of individual elements. I hope this "answer" is of interest anyway...

For affine hyperbolic curves, when $i$ and $i'$ agree on $K$, this happens only when $i$ and $i'$ are equal or complex conjugate of each other.

Let $f: \pi_1^{top}(X) \to \pi_1^{et}(X_{\mathbb C})$ be the natural dense inclusion, and $e_i,e_{i'} \pi_1^{et}(X_{\mathbb C}) \to \pi_1^{et}(X_{\overline{K}})$ be the natural isomorphisms defined by $i$ and $i'$. Let $\sigma$ be the element of the Galois group of $\overline{K}$ over $K$ that sends $i$ to $i'$, then $e_{i'}$ is $e_i$ composed with the action of $\sigma$ by outer automorphism of $\pi_1^{et}(X_{\overline{K}})$.

If the image of $e_i \circ f$ is conjugate to $e_{i'} \circ f$, then there must be some automorphism $\alpha$ of $\pi_1^{top}(X)$ such that $e_i \circ f \circ \alpha$ is conjugate to $e_{i'} \circ f = \sigma \circ e_i \circ f$ as a homomorphism. Now because $f$ is dense, we can extend $\alpha$ to an outer automorphism of $\pi_1^{et}(X_{\mathbb C})$ and thus to an outer automorphism of $\pi_1^{et}(X_{\overline{K}})$, and then the condition that these two maps are conjugate becomes the condition that these two outer automorphisms are equal in the outer automorphism group.

It is a result of Matsumoto and Tamagawa that this can only happen for the identity and comple conjugation (Mapping-Class-Group Action versus Galois Action on Profinite Fundamental Groups, Remark 2.1).

Furthermore, it looks to me that following their argument, the only case that the affineness assumption is used can be replaced with Theorem C(i) of Hoshi and Mochizuki