What is exact consistency strength of this kind of ZFC-Power set axiom theory? What is the exact consistency strength of $\sf ZFC - Power set $ plus adding a primitive total unary function $f$ and axiomatize that:
$\forall x \, \forall y: f(x)=f(y) \to x=y$
$\forall x \, \forall y: f(x)=y \to y \text{ is a von Neumann ordinal }$
In English: there exists a injective function from the universe to the ordinals.
And also axiomatize that: All sets are countable.
Is it equal to second order arithmetic?
 A: As you conjectured and Asaf confirmed, your theory is equiconsistent with Second-order arithmetic. Some textbooks even equate $\mathsf{ZFC^-}$ + 'every set is countable' with the second-order arithmetic! (e.g., A book by Cheng.) But I will assume that the second arithmetic is defined as we usually do.
Interpreting the second-order arithmetic inside $\mathsf{ZFC^-}$ is easy: just consider the model $(\omega,\mathcal{P}(\omega))$ as we do over $\mathsf{ZFC}$. Of course, $\mathcal{P}(\omega)$ is not a set in that case, but almost all part of the proof would carry over.
The main idea for the proof of interpreting $\mathsf{ZFC^-}$ with 'every set is countable' into the second-order arithmetic uses sets-as-tree construction.
You may find the concrete proof in Chapter VII of Simpson's Subsystems of Second Order Arithmetic. Let me briefly sketch its proof.
Work over the full second-order arithmetic. (In fact, $\mathsf{ATR_0}$ suffices for our definition: but it does not prove the full separation and replacement schemes.) A subset $X\subseteq \mathbb{N}^{<\omega}$ is a suitable tree if it is a nonempty tree (i.e., a subset that is closed under initial segments of finite sequences) which is well-founded. (That is, $X$ has no infinite branch.)
Now define the relation $=^*$ and $\in^*$ for suitable trees as follows:

*

*$S=^*T$ if and only if there is a tree isomorphism between $S$ and $T$. (The isomorphism must preserve the root.)

*$S\in^* T$ if and only if $S$ is isomorphic to a tree of the form
$$\{s\in\mathbb{N}^{<\omega} \mid n^\frown s \in T\}$$
for some $n\in\mathbb{N}$.

Now we can interpret set-theoretic formulas $\phi$ into that of second-order arithmetic $\phi^*$ by using $=^*$ and $\in^*$. Moreover, we have

Theorem. Let $\phi$ be a theorem of $\mathsf{ZFC}^-$ with 'every set is countable.' Then $\phi^*$ is a theorem of the second-order arithmetic.

(It follows from a combination of Lemma VII.3.20, Exercise VII.3.39, and Lemma VII.5.3. of Simpson's book.)

The above interpretation may not validate the global choice. However, this is easily resolved: the construction of $L$ carries over even without the axiom of power set, and it also validates the global choice.
