TortoiseGit with openssh key not authenticating using ssh-agent

Solution 1:

No tweaks needed.

Just make TortoiseGit point to the same ssh client used by git itself, see the screenshot:

This should be C:\Program Files\Git\usr\bin\ssh.exe in latest version of Git as mentioned by Aleksey Kontsevich in the comments.

Solution 2:

I first installed TortoiseGit with Plink and using Pageant to load ssh private key. The automatic authentication (Pageant) worked but setup was a fail as TortoiseGit don't recognize git repos url formatted as gitolite@gitbox/repo.git.

I finally found a workaround which consist to create a PuTTY session with the same name that the ssh alias (ie gitbox in the question).
This way I can clone as git clone gitbox/monrepo in the CLI and the origin syntax is correctly handled by TortoiseGit.