People who use screen readers to operate the Internet depend greatly on their familiarity with functions that appear on different pages within a site. Say, for instance, that you're once again making a poor financial decision online.
On one page, you find a printer icon which, when selected, prints your receipt of purchase. However, on another page on the same site, the printer icon prints the window. This would make the site significantly more difficult to use for anyone who relies on assistive technologies.
If components have the same functionality across a set of pages on your site, those components should be identified consistently. In this context, identification could mean a label, such as "search". In the case that it is an icon, such as in our printer example, or another non-text item, their alt text should be consistent across pages as well.
This rule only concerns consistency across pages. That being said, it is best practice for components with the same functionality to be identified consistently when appearing on the multiple times on the same page as well.