Your statement is equivalent to the assertion that there is a function choosing an enumeration of every countable ordinal. From an enumeration of $\alpha,$ you can easily inject it into a countable set of isolated points in Baire space. For the other direction, if $\varphi_{\alpha}$ injects $\alpha$ into Baire space, with the range a set of isolated points, there is a surjective partial map from basic open sets onto $\alpha,$ by sending $U \mapsto \beta$ if $\beta$ is unique such that $\varphi_{\alpha}(\beta) \in U.$ From this we can easily enumerate $\alpha.$ Note that by the Cantor-Bendixson analysis, any closed countable set of reals can be canonically enumerated, by sending basic open sets to the unique point in $U$ of maximal rank if there is such a point.
This principle implies there is an $\omega_1$-sequence of reals, since an enumeration of an ordinal can be canonically coded by a real. It is strictly stronger than the existence of an $\omega_1$-sequence of reals. For example, it clearly implies $\omega_1$ is regular, while in Figura's model ($\mathcal{M}36$ in Consequences of the Axiom of Choice), there is an $\omega_1$-sequence of reals yet $\omega_1$ is singular.