The custom object may not be displayed as a selectable item in App Manager because a Custom Tab has not been created.
Please check whether a custom tab has been created for the custom object. If not, try creating one by following the steps below.
Steps to create a custom tab
Click [Setup] from the gear icon in the upper-right corner of the screen.
In [Quick Find] at the upper-left of the screen, enter “Tabs” and click it.
In the [Custom Object Tabs] section, click [New].
Configure the following under [Object], then click [Next]:
Object: The relevant custom object
Tab Style: Any style of your choice
Select the profiles for which the tab should be visible, then click [Next].
Select the applications to which the tab should be included, then click [Save].
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