use mysql client which does not support authentication protocol can connect to tiproxy

Bug Report

Please answer these questions before submitting your issue. Thanks!

1. Minimal reproduce step (Required)

#  connect tidb directly
root@dbdeployer-0:/tools#  /root/opt/mysql/5.0.96/bin/mysql -u sha2 -h tc-tidb.testbed-endless-client-5lgf5 -P 4000 -psha --default-character-set utf8
ERROR 1251 (08004): Client does not support authentication protocol requested by server; consider upgrading MySQL client
root@dbdeployer-0:/tools#  /root/opt/mysql/5.1.72/bin/mysql -u sha2 -h tc-tidb.testbed-endless-client-5lgf5 -P 4000 -psha --default-character-set utf8
ERROR 1251 (08004): Client does not support authentication protocol requested by server; consider upgrading MySQL client

# connect to tiproxy
root@dbdeployer-0:/tools#  /root/opt/mysql/5.1.72/bin/mysql -u sha2 -h tc-tiproxy.testbed-endless-client-5lgf5 -P 6000 -psha --default-character-set utf8

root@dbdeployer-0:/tools#  /root/opt/mysql/5.0.96/bin/mysql -u sha2 -h tc-tiproxy.testbed-endless-client-5lgf5 -P 6000 -psha --default-character-set utf8

2. What did you expect to see? (Required)

3. What did you see instead (Required)

root@dbdeployer-0:/tools#  /root/opt/mysql/5.0.96/bin/mysql -u sha2 -h tc-tidb.testbed-endless-client-5lgf5 -P 4000 -psha --default-character-set utf8
ERROR 1251 (08004): Client does not support authentication protocol requested by server; consider upgrading MySQL client
root@dbdeployer-0:/tools#  /root/opt/mysql/5.1.72/bin/mysql -u sha2 -h tc-tidb.testbed-endless-client-5lgf5 -P 4000 -psha --default-character-set utf8
ERROR 1251 (08004): Client does not support authentication protocol requested by server; consider upgrading MySQL client
root@dbdeployer-0:/tools#  /root/opt/mysql/5.1.72/bin/mysql -u sha2 -h tc-tiproxy.testbed-endless-client-5lgf5 -P 6000 -psha --default-character-set utf8
Welcome to the MySQL monitor.  Commands end with ; or \g.
Your MySQL connection id is 100
Server version: 5.7.25-TiDB-v7.3.0-alpha

Copyright (c) 2000, 2013, Oracle and/or its affiliates. All rights reserved.

Oracle is a registered trademark of Oracle Corporation and/or its
affiliates. Other names may be trademarks of their respective
owners.

Type 'help;' or '\h' for help. Type '\c' to clear the current input statement.

mysql> select tidb_version();
ERROR 2006 (HY000): MySQL server has gone away
No connection. Trying to reconnect...
Connection id:    100
Current database: *** NONE ***

ERROR 2013 (HY000): Lost connection to MySQL server during query
mysql> exit
Bye
root@dbdeployer-0:/tools#  /root/opt/mysql/5.0.96/bin/mysql -u sha2 -h tc-tiproxy.testbed-endless-client-5lgf5 -P 6000 -psha --default-character-set utf8
Welcome to the MySQL monitor.  Commands end with ; or \g.
Your MySQL connection id is 100
Server version: 5.7.25-TiDB-v7.3.0-alpha

Copyright (c) 2000, 2011, Oracle and/or its affiliates. All rights reserved.

Oracle is a registered trademark of Oracle Corporation and/or its
affiliates. Other names may be trademarks of their respective
owners.

Type 'help;' or '\h' for help. Type '\c' to clear the current input statement.

mysql> select tidb_version();
ERROR 2006 (HY000): MySQL server has gone away
No connection. Trying to reconnect...
Connection id:    100
Current database: *** NONE ***

ERROR 2013 (HY000): Lost connection to MySQL server during query
mysql> exit
Bye
root@dbdeployer-0:/tools#  /root/opt/mysql/5.0.96/bin/mysql -u sha2 -h tc-tiproxy.testbed-endless-client-5lgf5 -P 6000 -psha --default-character-set utf8
Welcome to the MySQL monitor.  Commands end with ; or \g.
Your MySQL connection id is 100
Server version: 5.7.25-TiDB-v7.3.0-alpha

Copyright (c) 2000, 2011, Oracle and/or its affiliates. All rights reserved.

Oracle is a registered trademark of Oracle Corporation and/or its
affiliates. Other names may be trademarks of their respective
owners.

Type 'help;' or '\h' for help. Type '\c' to clear the current input statement.

mysql> \s
--------------
/root/opt/mysql/5.0.96/bin/mysql  Ver 14.12 Distrib 5.0.96, for unknown-linux-gnu (x86_64) using readline 5.1

Connection id:          100
SSL:                    Not in use
Current pager:          stdout
Using outfile:          ''
Using delimiter:        ;
Server version:         5.7.25-TiDB-v7.3.0-alpha
Protocol version:       10
Connection:             tc-tiproxy.testbed-endless-client-5lgf5 via TCP/IP
Client characterset:    utf8
Server characterset:    utf8
TCP port:               6000
--------------

[2023/07/17 08:38:10.524 +00:00] [INFO] [main.proxy] [proxy/proxy.go:170] [new connection] [connID=374] [client_addr=10.233.105.34:46090]
[2023/07/17 08:38:10.524 +00:00] [WARN] [main.proxy.conn.be.authenticator] [backend/authenticator.go:148] [frontend may not support plugin auth] [connID=374] [client_addr=10.233.105.34:46090] [capability=CLIENT_LONG_PASSWORD|CLIENT_LONG_FLAG|CLIENT_LOCAL_FILES|CLIENT_PROTOCOL_41|CLIENT_INTERACTIVE|CLIENT_TRANSACTIONS|CLIENT_SECURE_CONNECTION|CLIENT_MULTI_STATEMENTS|CLIENT_MULTI_RESULTS]
[2023/07/17 08:38:10.526 +00:00] [INFO] [main.proxy.conn.be] [backend/backend_conn_mgr.go:229] [connected to backend] [connID=374] [client_addr=10.233.105.34:46090] [ns=default] [backend_addr=tc-tidb-0.tc-tidb-peer.testbed-endless-client-5lgf5.svc:4000]
[2023/07/17 08:38:10.568 +00:00] [INFO] [main.proxy] [proxy/proxy.go:181] [connection closed] [connID=374] [client_addr=10.233.105.34:46090]
[2023/07/17 08:38:24.628 +00:00] [INFO] [main.proxy] [proxy/proxy.go:170] [new connection] [connID=375] [client_addr=10.233.105.34:46146]
[2023/07/17 08:38:24.629 +00:00] [WARN] [main.proxy.conn.be.authenticator] [backend/authenticator.go:148] [frontend may not support plugin auth] [connID=375] [client_addr=10.233.105.34:46146] [capability=CLIENT_LONG_PASSWORD|CLIENT_LONG_FLAG|CLIENT_LOCAL_FILES|CLIENT_PROTOCOL_41|CLIENT_INTERACTIVE|CLIENT_TRANSACTIONS|CLIENT_SECURE_CONNECTION|CLIENT_MULTI_STATEMENTS|CLIENT_MULTI_RESULTS]
[2023/07/17 08:38:24.631 +00:00] [INFO] [main.proxy.conn.be] [backend/backend_conn_mgr.go:229] [connected to backend] [connID=375] [client_addr=10.233.105.34:46146] [ns=default] [backend_addr=tc-tidb-0.tc-tidb-peer.testbed-endless-client-5lgf5.svc:4000]
[2023/07/17 08:38:24.658 +00:00] [INFO] [main.proxy] [proxy/proxy.go:181] [connection closed] [connID=375] [client_addr=10.233.105.34:46146]

4. What is your version? (Required)

Start with the warning at backend/authenticator.go:148 and the backend connection path at backend/backend_conn_mgr.go:229. Reproduce the behavior using the MySQL 5.0.96 and 5.1.72 clients against TiProxy, then compare it with direct TiDB connections. Done means the reported client compatibility behavior is understood and the connection remains usable for the demonstrated query, with regression coverage where the existing tests belong.