Do you have significant ownership in the company? If not, I don't see any reason for you to give frank feedback.
You can't know for sure whether the information will reach them, nor whether they will be able to infer who provided the feedback.
The best possible outcome for you if you have no ownership is...senior management is better informed on what employees think of this manager. You don't know what they'll do with this info.
The worst possible outcome for you is that the manager is able to infer it was you that provided certain feedback, and that they are upset by this feedback. This could result in you losing the job eventually (they could make your life difficult, claim you're not performing well etc. etc.)
It probably makes the most sense (assuming you're primarily concerned with remaining employed to provide for your family) to maximize the level of positivity in all of your feedback.
You can't know for sure whether the information will reach them, nor whether they will be able to infer who provided the feedback.
The best possible outcome for you if you have no ownership is...senior management is better informed on what employees think of this manager. You don't know what they'll do with this info.
The worst possible outcome for you is that the manager is able to infer it was you that provided certain feedback, and that they are upset by this feedback. This could result in you losing the job eventually (they could make your life difficult, claim you're not performing well etc. etc.)
It probably makes the most sense (assuming you're primarily concerned with remaining employed to provide for your family) to maximize the level of positivity in all of your feedback.