I find SSR has now become a ambiguous term.
SSR as in React SSR, the server-side rendering of single page apps to improve load time and SEO: Flarum is not using that.
SSR as in simple server-side generated HTML, Flarum is using that to generate the noscript view (via Laravel Blade templates)
Flarum has both a noscript HTML template, and the single page app that runs client side.
I'm not sure what each search engine sees, but they will see one or the other. From what I understand Google does run javascript, so they are probably able to read the page as it looks from the single page app (like in any javascript-enabled browser). Other "less advanced" search engines will see the noscript version.
Everything header and meta-related is the same in both cases as it's part of the base HTML of the page, whether the search engine can read javascript or not.