ssh login to github

The problem is, you are trying to use the public key for authentication. The private key, is e.g. id_rsa2 while the public key is id_rsa2.pub, at least per default, when generated with ssh-keygen.

As default ssh uses id_rsa for identification, so you have to use:

ssh -i ~/.ssh/id_rsa2 -T git@github.com

The ssh config should look like this:

Host github.com
HostName github.com
User git
IdentityFile ~/.ssh/id_rsa2

The Host value is important, because that's what you have to use to use the configuration. You now can use ssh github.com. If you don't change the Host line, you need to use ssh github, but this could break other tools.

Alternatively you could use ssh-agent. In this case you do not need to create the config and just add the key to your ssh-agent and ssh to github.

ssh-add ~/.ssh/id_rsa2
ssh -T git@github.com