I came across a similar issue while trying to dump my 5.1 database for upgrade to version 8.

The quick and dirty way to get things to work is to just create a user on your 5.1 server with no password, and bind that user to just one IP address. Delete that user when you're done.

Assuming your client IP is 10.21.1.1:

CREATE USER 'nopass_user'@'10.21.1.1' IDENTIFIED BY '';
GRANT SELECT, INSERT ON *.* TO 'nopass_user'@'10.21.1.1';
FLUSH PRIVILEGES;

Version 8.0.35 was supposed to check and allow for the older handshake but that didn't work for me. On another thread a developer said that they won't support ancient versions so I wouldn't expect any proper way of getting things to connect.