I expected that this only happens via cookies (and other locally stored state) combined with login accounts since IP addresses are commonly shared, including between parents and children, and your ISP can reallocate them, but see the Update below.
You can protect from the IP address possibility by using your phone's cellular connection (turn off WiFi), or using a cafe or library WiFi (if they run WiFi even when closed during a pandemic), or a VPN (if your partner isn't also using it, as would happen if your router handles VPN for the entire LAN), or an anonymizer service.
Facebook reportedly even tracks people who don't have accounts, so start by clearing all Facebook cookies, clearing the browser cache, and restarting the browser. There's also a Firefox extension Facebook Container that makes it harder for Facebook to use third-party cookies to track your visits to other websites.
Perhaps your browser is logged into one of your partner's accounts or vice versa. The easiest workaround is to use Firefox private browsing mode (or Chrome/Chromium incognito mode). That begins with no cookies and deletes them afterwards.
Be careful about logging into an account while doing this, say Amazon, since that account can track your interests. If you need to log in, consider creating a fresh account or making sure that account isn't logged in on your partner's computer or linked to other accounts.
You can use duckduckgo.com for web searches that never track you. That step is good for web searches but the commerce sites you reach might still track you via cookies, so definitely use private browsing mode.
A way to test these measures is to search for some unusual and different products and services like audio mixers, robot arms, pickup trucks, skis, house painting, estate lawyer, or car repair. See if that "interest" crosses over to your partner's computer.
After product searches on Amazon, I've seen ads on other websites for those specific products, not just the categories. Maybe that happens via Amazon cookies or via IP address.