It's not so much that info is stored on the router, imagine receiving a letter, the house number and postcode is akin to your public IP address telling the posty where to deliver the letter, whoever picks up the delivered mail is akin to the router, dishing out the letters to their intended recipients. Poorly described analogy aside, essentially most households connect to the Web using a single IP and that IP could well be associated with looking at/being interested in xyz.
Another possibility would be that your daughter logged onto her FB using your device, Facebook may then associate your 'browsing history' with her account. Or even if you and your daughter are friends on FB and if, for example, she's liked a photo of you fishing or similar, FB 'relates' this information and guesses your daughter likes fishing so may be interested in the same fishing ads that you may be interested in.
As for 'adult stuff' I think it would depend on exactly what, for example if you were looking at buying ToH something then I'd imagine that could well pop up in an advert. This is where parental controls and the like come in, NSPCC have some good information on this and more https://www.nspcc.org.uk/preventing-abuse/keeping-children-safe/online-safety/parental-controls/
Close to digressing so to come back on topic, there are lots of ways that sites can record or 'best guess' your online activity. If you don't want to be tracked the only thing you could do is turn off javascript, flash and cookies, but your online experience will significantly decrease. Whitelisting sites as needed is a possibility but soon becomes a pain in the a**e. And, this doesn't stop your ISP or the sites you visit from logging 'your' (as in IP address') activity, with the potential that the logged information is shared.
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