Hello @Nina244,
When I check my own listing it appears that it's as @Matthew285 says. You should make sure that you're logged out of your Airbnb account when you search for your listing.
When I go to airbnb.com and find my own listing in Limerick in Ireland it has "edit listing" in brackets as you refer to. You can see that, and the profile photo in the top right meaning that I'm logged in, in the first screen shot below.
When I'm logged out I don't see my profile photo or the "edit listing" link. That's shown in the second screen shot below.
To log out, click on your profile photo and then select "Log out" from the menu.
The screen shot below is when logged out and it doesn't have the "Edit listing" link.