I created a new user with default login shell (bash) and renamed fzf.bash to fzf.sh. For some reason, fzf is not active on startup, but if I source it manually, or if I resource /etc/profile it works.
The OP claims it works. One of us is making a mistake.

Anyhow, I believe that fzf.bash and fzf.zsh should not belong to /etc/profile.d which is meant to be generic and sourced by different shells. /etc/profile.d is usually needed to update PATH automatically.
I'd rather move fzf.bash and fzf.zsh to their respective /usr/share folders.

What would be those folders by the way? There is /usr/share/zsh, but no "keybindings" subfolder. I do not know any such folder for bash.

Ambrevar, are you using a display manager? Some of them don't source /etc/profile, or don't source it correctly.

You are right, there is an issue with the LXDM since it works when I connect from a TTY, but still: the PATH get sets properly (perl, plan9) when started from LXDM, which means /etc/profile gets sourced, as /etc/lxdm/Xsession testifies.
One clue: this Xsession script is a "/bin/sh" script, while fzf.bash calls for the full bash capabilities. Could this be the issue? I cannot test it just now, will do later.

This is by design. You should source that file manually from there. We don't want to add intrusive stuff like fzf automatically to your shell. Everything is as it should be. However, I do agree with the reasoning that this stuff is better moved somewhere else. I'll move it to/usr/share.

Move which file where? What is by design?

Pierre, please check your mail.

Well this seems fixed?

Indeed.