I have forgotten my Facebook account password and also I have no access to the number which I use to log-in to Facebook account.
On this page (see the image below) they are asking for a government photo-ID to verify my account. The problem is that I don't have a driving license, voter ID card, passport etc. All I have is a school ID card but that hasn't worked for me. Is there any other way to recover my password?
It is not possible to recover an account without a government issued ID. You can recover your account using your email account. If you don't have access to your email and phone number you need to provide a government issued ID at any cost to verify that it belongs to you legally. From my personal experience, I provided government issued ID but I was not able to recover my account after that.
Normally one would have access to the email account from setting up the account in the first place... barring that, if you do not currently have a photo ID, it should not be impossible for you to get one. They do not indicate that it must be an ID that you had prior to account creation ;).