bashrc not loaded in /bin/bash shell

the ~/.bashrc file is loaded for interactive non-login shells, that's why running /bin/bash from ssh works as you expect.

For login shells (amongst others) ~/.bash_profile is read. Typically your ~/.bash_profile contains something like

if [ -f ~/.bashrc ]; then
        . ~/.bashrc
fi

Which ensures that ~/.bashrc is loaded if your shell is a login shell.

If your ~/.bash_profile contains the above and ~/.bashrc is not being loaded you can debug the scripts by using set -x as usual.

Normally when a user is created it will copy a bunch of files to your new homedirectory (if you chose to create one).

You can check the /etc/skel directory for those files. There should be the .bash_profile that you expected.

As none of the previous, it seemd that the user didn't have a shell assigned, so it just show the character $ as prompt.
I had to assign a shell to the user by running the command:

sudo usermod --shell /bin/bash username

After that, the user automatically logged in with a bash shell.