Get rid of Messages' notifications when application is connected to a PC
For disabling first notifications (i.e. the one that appears on the phone when it is connected to PC) one must:


tap such notification (or select Messages for web from three-dot menu in Messages application),
tap three-dots menu and select Notifications and
uncheck Show presistent notification option.




As for disabling second notifications (i.e. the one appears in PC / Chrome / web browser version of Messages for web, about phone using mobile data) this is most likely solved in the Google-way.

Thus there is no way to disable this because Google thinks that you must see this and doesn't actually care what you really need or want, sorry.