So I have a web part page, where I added a list view web part. Though I have no option to export the list items to excel. I don't even have the 'List' and/or 'Item' tabs on the ribbon if I check any of the items. Is this because it's a web part? Can I somehow manage to export the content of the list?
No. A list is always presented in the same way, using the same web part either if you go to the original list location or add it somewhere else. The only difference there is for your web part page is that the list web part toolbar type has been set to
If you choose to edit the page and then edit the list web part, you can change the setting
Toolbar Type to
Full Toolbar or
Summary Toolbar. Apply the changes to the web part and save the page.
You should now be able to click anywhere on the list web part zone and the toolbar with the "Item" and "List" option will be available and from here you can do anything you want with the list again, including exporting to Excel. If you click outside the list web part zone the toolbar should disappear again.
Yes, it is because you are looking at the Items as through a window
If you can see the items in a WebPartPage you have access to the List
Open the original List, usually the Title of the WebPart leads to the List.
Or find the List in Site Contents (left Quick Launch menu or top right cog icon)