Similar theme like Mesmerize

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Why can’t you use the same theme?
Since you are a beginner, I’ll tell you that this type of theme is not a good idea, since it puts so much of the home page content into theme options. That content should be in the Page itself, not in the theme. If you switched themes, your home page would be very different. You should be able to switch themes at any time, because sites get hacked and the theme is often targeted since it is called for each page request.
Making a change on the home page should check the capability edit_page, not edit_theme_options.

Right, you could use the same theme. You can customize a theme’s appearance a lot with just CSS added to the Additional CSS panel. For changes that are more extensive which require custom PHP code, we normally create custom plugins. But if you find the need to alter your theme’s template files, you should opt for the child theme approach. Watch out though, some themes are already child themes. You cannot create a grandchild theme. If your theme’s style.css file has a comment header that includes a Template: line, it’s already a child theme.

It’s possible to use custom templates from a plugin, but it’s a more advanced technique. Only do so if there is no other choice.

I’m not sure what you have in mind when you say you want “an option manage page section” for a second site. For the standard WP installation, the answer is no. You can convert your WP installation into a multisite network, but if you will be limited to two sites I personally would consider a second single site installation over multisite. WP networks have a lot of quirks which I’d just as soon avoid if I’m only adding one more site.