Yes, Google does decide on their own what your web page is about, and they generally do not pay attention to any attempts by webmasters to manipulate the keyword relevance outside of what users can expect to find when they arrive at your page.
One of the reasons the founders of Google launched their search engines was because they noticed that many webmasters were manipulating other search engine results by abusing the meta keyword tag and keyword stuffing their web pages with irrelevant keywords. So, yes Google was founded based on the assumption that not all webmasters can be trusted, particularly when it comes to the competitiveness of ranking in search results.
It seems that you have a fundamental misunderstanding of what a search engine is and what it does. You are right, search engines do not care what your “website” is about, because search engines do not index “websites”, they index individual web pages. Web apps that list “websites” are called web directories and they serve a different function from a search engine.
Try to keep in mind that there are often many pages that are relevant to a keyword term so search engines, like Google, try to bring the most relevant and useful content to the top and ignore less relevant and less useful content. Google doesn’t decide completely on their own about which page is useful and which are not. They also analyze user behavior to verify and adjust the core results.
So, if you have a page that isn’t doing well, in search ranking then it is because Google or Google’s search users have found your page lacking as compared to other pages, for the keyword search term.
To learn more about how Google uses keywords to index and rank your web pages read Google’s guide for search engine optimization
Google found a better approach, and that was to ignore the mostly useless meta tags and to decide on their own what a web page is about, based on the content, as rendered, and through examining content on the web of documents that connect to the page (PageRank).