So you're asking if trying to rank --ex. www,website,com/subject will be the same as trying to rank www,subject,com?

Yes, it's the same. The domain doesn't matter.

Yes, I am asking exactly the same. So actually the exact-match domain doesn't really matter?

So where I should build my backlinks?
On the page - www.website.com/subject
Or on the whole website - www.website.com ?

Even exact-match domains and web URL structures today may not seem to be that functional from web marketing or SEO perspectives. If this was the case, webmasters would be simply purchasing keyword-rich domains, registering tens of them and mastering the niches online. As you have already heard, at present, links are playing crucial roles over the net, affecting web ranks, even if your domain apparently is not related to the financial niches and sectors it is involved in.

You can find a lot of top-ranking domains on Google, even the ones which never contain any terms related to the niches targeted and still having superior ranks due to their strong incoming links, powered by the primary anchors. Working on mini sites and registering several domains and redirecting all of them to just one single portal was effective several years ago, or maybe it still helps with getting improved ranks on Bing and Yahoo engines but the Google case can be different.