It may be the internet browser on your Android phone blocking the page from opening the login section as a new "window". To enable this function, can you try the following:
- Open Chrome on your phone
- Press the "Settings" button on the top-right of the screen. This button looks like three vertical buttons.
- Press "Settings"
- Press "Site Settings"
- Press "Pop-ups and redirects"
- Toggle the on/off button on the top-right of the screen to alter this.
Following this, re-launch the Bose Music app, and attempt to connect your Spotify account once again?
You posted initially about the ability to play audio to the speaker via the Spotify app - it sounds like you will be using Spotify Connect for this functionality, not necessarily using the Bose Music app.