Facebook Marketplace search function disabled

What generation iPhone and what version of Safari?

I just checked it on my iPhone 13, and I can still search Marketplace with the search function on Safari. The app doesn't really let you open multiple tabs simultaneously but it's easy to save ads to go back to them.

I've never had any issue like you describe with apps, and internet browsers all collect cookies unless you disable those functions. Which you can do with apps as well, BTW.