For 7.2 and prior versions, the OOTB themes append the instance name in their templates. See, for example, portal_normal.ftl in frontend-theme-classic.
The format is: Page Title - Site Name - Instance Name
It cannot be changed from settings, although there is a feature request LPS-181060 "Aligned with the Roadmap".
If you want to change this, it can be achieved through customization.
Even if you remove Site and Instance names from the full page title (i. e., from the <title> tag of the page) they can still be appended after the page title in search results pages. And they can suffer other changes as well.
Why is that? Because the search engines have the final word about how to display their search results in order to improve the user experience.
Although they take into consideration the <title> tag to provide their results, they take into account many more things too, such as the browser cache and history, logged-in sessions, geographic location, user's language and preferences, content of the page and its relevance, hyperlinks, sitemap, etc.
Therefore, they can append the site/company/institution name in order to help the user quickly identify the website, for instance, if it is a popular brand/entity or the user includes the company name in the keywords.
