I think the 404 Component Not Found on the frontend might be related to https://docs.joomla.org/J3.7.1:Non_SEF_Urls_Broken

I cannot replicate the backend issue

I have noticed this once, but also agree it shouldn't be an edit button since you cannot edit it, only select another.

Really it should just be a single select to select another item.

Just noticed this on a 2nd site, after the Update the wrong URL looked something like
/?Itemid=169
After fixing it manually the URL was a SEF URL such as
/tutorials
for example.

I have noticed this once, but also agree it shouldn't be an edit button since you cannot edit it, only select another.

The "Edit" button should only appear when there is a menu item selected (ie, with the id in jform_id form field).
See https://github.com/joomla/joomla-cms/blob/staging/media/system/js/modal-fields-uncompressed.js#L36

So that means that in that particular case that jform_id form field (hidden field) must have a menu item id and that's why you have the "Edit" button.

Now, why the title of the menu item is not displayed as is displayed "Select a Menu item"...

Could it be possible that you have a menu item alias to a menu item that existed in the past but doesn't exist now?

To help anyone further debbuging this issue, the menu item title is fetched here: https://github.com/joomla/joomla-cms/blob/staging/administrator/components/com_menus/models/fields/modal/menu.php#L201

It has been several months since this issue was created without confirmation of the issue. I am closing it at this time but it can always be reopened