The persistence of the Paper Tape window's contents depends on a system-wide setting.
The setting is controlled by:
System Preferences > General > checkbox: Close windows when quitting an app
When the checkbox is on, Calculator (and many other apps) will automatically preserve the location, visibility, and contents of windows between launches.
When the checkbox is off, apps will typically discard (or lose) either location, visibility, or contents of windows when the app is quit.
Look at the state of this checkbox on each computer, and see what it says.
This may be coupled to another checkbox: Ask to keep changes when closing documents
Whatever the checkbox's state is, you can "reverse" it for a given app by holding down the Option key when choosing Quit from the app's menu. The menu item changes from Quit APP_NAME_HERE to one of:
Quit APP_NAME_HERE and Close All Windows
Quit APP_NAME_HERE and Keep Windows
For example, if your checkbox is normally checked, you can preserve the Paper Tape in Calculator by quitting with cmd-option-Q instead of cmd-Q.