I think to give sb a shake is like to give sb a fair shake - an American informal idiom: a fair treatment that gives you the same chance as sb else.
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give someone a shake - (coll.) 1. shake a sleeper to rouse him: She thought he might fall asleep standing there and grabbed his arm to give him a shake.
2. rouse a person to action: Danny was not playing as well as he is now at the start of the season. We needed some way to give him a shake and he has responded really well.
3. make smb. suffer a nervous shock: It gave me such a shake a