Assign a custom name to the Emacs frame (so that WM rules can apply to it)

[Kabouik]

In my Sway configuration, I use the following to open a floating window/frame for Emacs:

for_window [title="^floating"] floating enable
bindsym $mod+mod1+Ctrl+e exec emacsclient -c -a "" -e '(raise-frame)' -F '((name . "floating emacsclient"))'

I would like Atomic-Chrome/GhostText to similarly assign the "floating" keywoard to the Emacs frame they open, so that Sway automatically sets it to floating mode. How should I proceed? Below is my current configuration for Atomic-Chrome but I am not sure how to change the default name of the frames it opens:

(require 'atomic-chrome)
(atomic-chrome-start-server)
(setq atomic-chrome-default-major-mode 'markdown-mode)
(setq atomic-chrome-extension-type-list '(ghost-text))
(setq atomic-chrome-buffer-open-style 'frame)

I would also like it to start an emacsclient frame by default, I don't know if that is already the case. Thanks for the help!