I am not an accountant. I would suggest you consult an accountant.
Can you clarify, is this the account that you and your partner sell from?
Ebay will report your sales from this account to HMRC. You have reached the threshold of 30 items so they are obliged to report your sales. If you don't give them your NINO, they will still report your sales, along with your full name, home address and bank account details. Realistically, even if you don't give Ebay your NINO, it will take HMRC minutes to find it.
So, you may as well give them your NINO.
As far as your Ebay related questions go, your partner can't take over your account. The National Insurance Number has to belong to the account holder.
Personally, I would do the following things:
Add my NINO to the account.
Stop selling items bought for profit on this account immediately. Your partner should have his own business account for selling such items. This will help with book-keeping and tax returns.
Use this account only for selling personal, unwanted items.
Print off details of sales for this year and possibly last year and divide sales into 'for profit sales' and personal item sales, just in case HMRC do get in touch.
"There are two ways to be fooled. One is to believe what isn't true; the other is to refuse to believe what is true.”
Søren Kierkegaard, Danish philosopher (1813 - 1855)