Manage or turn off your alerts
Saved searches are managed one by one at /account/searches. Everything else has a switch in Settings, Notifications.
Turn off one saved search
Open /account/searches and delete the search you no longer want. Deleting it stops its alerts. There is no separate mute, because a saved search exists in order to alert you.
Turn off a whole type of alert
Go to Settings, Notifications. Each kind of alert has its own row, and email and push are independent switches, so you can keep push and drop email or the other way round. The ones relevant here are search alerts, price drops, and the alerts about listings you posted.
Unsubscribe from an email you are holding
Search alert and price drop emails both carry an unsubscribe link at the bottom, and both support your mail app's own one-click unsubscribe button. Using either turns off that kind of email for your account. It does not delete your saved searches or your liked items, and it does not affect messages from buyers and sellers.
Quiet hours
If push notifications are the problem rather than the volume, Settings has quiet hours: push is muted during the hours you set, and email still goes through.
What you cannot turn off
Messages from another member about a listing, and the transactional email that follows something you did, such as a receipt. Those are how the marketplace works rather than marketing. If somebody is contacting you and you want it to stop, block them from the conversation instead.
Still getting email after switching everything off
Check that you are signed in as the account the email is addressed to. People often have two: one from Google or Apple sign-in and one from an emailed sign-in link. Alerts follow the account, so switches set on one do not apply to the other.