Based on the image you posted on your questions comments, I would assume the Videos, Blog, and Photos are custom post types registered by your current theme. The theme might also register some custom taxonomies for the post types. The post types and taxonomies could also be registered by some theme related plugin.
You should either take a look at the theme’s code or see the browsers address field when viewing the Videos, Blog or Photos admin pages to see what names have been used to register the custom post types. The same goes with the any custom taxonomies. You could for example see something like &post_type=videos.
Then just create a custom plugin in which you register the custom post types and taxonomies with the same names to use them after theme change.
Create plugin,
https://developer.wordpress.org/plugins/intro/
Register post types,
https://developer.wordpress.org/plugins/post-types/registering-custom-post-types/
Register taxonomies,
https://developer.wordpress.org/plugins/taxonomies/working-with-custom-taxonomies/
This is what Gopalakrishnan18 was also saying in the comments.