Can client add/share Stock subscription to my Adobe CC account?

Yes, that is possible. However, there are some cautions. First, you will have a Teams account (like having a company's office and a home office, you can't be in both at the same time. @Test Screen Name is correct in that sense).

Also: you will be able to licence assets for their use, but you can't licence for your use. They can't do a business from that. BTW: a stock subscription works differently than a CC subscription. You have one plan (like the 750 assets/month) and everyone draws from that plan. So basically, there can be an infinite number of users. And as I said, you will be limited to licence for their use or for work, you do on their behalf.

The interface between them (your account with them) and you (your “private” account) will be more complex than that. I do not know if you can share data between them, using shared libraries, I have never done that.

I see no problem trying that out. When the customer withdraws the account, you won't be able any more to log in to your (their) business account. Your own account will stay unharmed.

You should check your options with the administrator of your customer's Teams subscription.

They can set you up an account and you can use it (according to their terms). But it's a separate account, even if the emails are the same. You cannot be logged into their account and your account at the same time. (Well in theory you could have the local desktop with one account, and a web browser licensing stock on another. But purchases will go into the cloud account of the web user, and so won't be available on the desktop. Best to avoid even trying this)