Does Tails "leak" hardware serial numbers?

Generally, no.

Tails "leaks" MAC address on the local network, unless https://tails.boum.org/doc/advanced_topics/mac_changer is used.

In the sense that whenever you visit a website, your information gets logged into some sort of database?

Some information will likely be logged, but it should not contain hardware serial numbers.

What about other operating systems?

This is off-topic here, sorry.