Like some others who have responded, I welcome it when a Project becomes a manager of a profile I have created or managed myself. I do not feel elbowed out in any way, nor am I upset if there is no prior consultation.
I would expect Projects to put explanatory comments on profiles when they add themselves as profile managers or add Project protection.
There are times when Project protection needs to be added fast to safeguard the integrity of information on WikiTree. Project protection requires a Project to become a profile manager.
In other circumstances, where quick protection is not needed, how far it may be appropriate to consult existing managers, as opposed to just notifying them, depends partly on the nature of the Project.
Looking at the Projects I lead, I would not expect a Project like the Magna Carta Project to seek permission from profile managers before adding itself as a manager: the nature of the Project means that it seeks to manage profiles in lines of descent from Magna Carta barons to Gateway Ancestors. The Project would, though, put an explanatory comment on the profile. There are other Projects in a similar position.
For the Quakers Project, on the other hand, where there were existing active profile managers I would put a comment on the profile saying I would like to add the Project as a manager, explaining why, and then give the existing managers a little time to respond or object (I have never yet had an objection). If the existing profile managers have been inactive for a considerable time, I might not do this, instead just adding the Project as manager with an explanation.
My general experience is that usually profile managers are pleased when Projects - including Projects I lead - seek to co-manage a profile.