That's a tough one, by design the specific configuration for a widget is persisted for the page/template the widget is dropped on, so in the case where a widget is dropped on a page template, all pages using this template will inherit the widget and its configuration from the template.
For the specific use case scenario you have outlined we would advise for either filtering the displayed videos by category, depending on the page, as you properly pointed out, or dropping a new widget and setting its sepcific configuration on every page.
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