Is my spouse's savings account joint because we share a checking account?

An account in only one spouse's name is that spouse's account. The other spouse has a financial interest in it only if it is held for them by someone acting as their agent or nominee. They have signature authority over it if they can instruct the bank to pay money out of it.⁠^{[1]Source 1}⁠^{[2]Source 2}

You and your spouse share a foreign checking account. Your spouse also has a savings account in their sole name that transfers money to the checking account. The transfers don't make the savings account joint.⁠^{[1]Source 1}⁠^{[2]Source 2}

Check the account agreement, named owners and bank mandate for the savings account. Report each account on its own ownership and authority facts.⁠^{[2]Source 2}