When a lender refers to getting a mortgage "off their books" it means they are going to sell your mortgage to another lender or servicer. Lenders refer to selling a mortgage as getting it "off their books" because the loan is no longer on the lender's financial books or balance sheet.
It is very common for lenders to sell mortgages and the decision to sell a mortgage is not related to the borrower or the type of loan. In short, selling a mortgage is a financial decision by the lender that enables them to free-up capital to provide more mortgages to more borrowers and make more money.
Lenders sell loans all the time and I had a similar experience to you when my mortgage was sold to a different lender soon after my loan closed. The only thing that changed is that I started making my mortgage payment to a different company.
From a borrower's standpoint, when a lender sells your mortgage nothing really changes except for you make your monthly payment to the company that bought your loan. Your loan terms such as your interest rate, mortgage length, loan features and monthly payment all remain the same. Additionally, you cannot be charged any extra fees when your mortgage is sold.
It is important to highlight that when the lender sells your mortgage it does not interfere with or change the ownership of your home in any way. You still own your home as reflected by the grant deed that was recorded when your loan closed and you bought your home.
What is a little unusual about your situation is that your current lender is requesting a copy of your social security card at this point in the process. Because your loan just closed, the mortgage company should have all the documents required to sell your loan and should not make any more information requests. Furthermore, unless you obtained an FHA loan, you typically do not need to provide a copy of your social security card to your mortgage lender.
I recommend that you contact the individual who originated and processed your loan at your mortgage company to confirm that the request is not a scam. Contacting your lender also should enable you to understand why the company needs your social security card or any additional information after your loan has closed. Because the circumstances you outlined are somewhat atypical, speaking directly to your point of contact should alleviate any potential concerns.