Ok, talked to the Sway folks and they confirm that Sway does not implement the Settings interface and that they don’t plan to do it.
IIUC, the following configuration should use the GTK portal as fallback:
$ cat /usr/share/xdg-desktop-portal/portals/gtk.portal
[portal]
DBusName=org.freedesktop.impl.portal.desktop.gtk
Interfaces=org.freedesktop.impl.portal.FileChooser;org.freedesktop.impl.portal.AppChooser;org.freedesktop.impl.portal.Print;org.freedesktop.impl.portal.Notification;org.freedesktop.impl.portal.Inhibit;org.freedesktop.impl.portal.Access;org.freedesktop.impl.portal.Account;org.freedesktop.impl.portal.Email;org.freedesktop.impl.portal.DynamicLauncher;org.freedesktop.impl.portal.Lockdown;org.freedesktop.impl.portal.Settings;
UseIn=gnome
$ cat ~/.config/xdg-desktop-portal/sway-portals.conf
[preferred]
default=wlr;gtk;
But for some reason it doesn’t 