How to keep track of data from previous site when signing into another [closed]

Apologies if this question is very rudimentary, but I don't really know where to start, so let me explain my use case.

I am searching on a website1 for a specific product or service.

The website1 I am on doesn't have it, but it links me to another website, website2, that does.

The other website2 that does, needs me to log in before I can access it.

I want to pass the query parameters, product data, service data, to Website2 (basically so I don't need to search again).

Is this possible? Can anyone explain to me what methods, or functions they would use for this?

As a follow up, if this data can be passed, can it survive a situation where my log in on website 2 fails due to bad credentials.

Thanks a lot.