There's an option to install the "bash" applet as a link to either ash or hush, the two shells that busybox comes with. Turns out that a large number of "bash scripts" use no bash-specific features in spite of using "bash" in the #!, or only a few bash-specific features like [[ ]].

It's disabled by default, and arguably not a good idea to enable it because compatibility is not great as you found out. Either way, "busybox bash" doesn't exist: only the option to alias ash or hush to bash.