We are using a multisite install with Ajax Search Pro and when I go to Front End Search Settings > Category & Taxonomy Terms > Change Display Mode > alter the text display of the menu item, this changes on both sites. Everything about these sites are completely separate from one another but this doesn’t seem right. They should be independent of one another, not sharing things like this text which presumably people who change for language purposes like our case.
See below. The taxonomies are correct but not the text display of the select menus.
The search instances are shared across wordpress Multisite Installations on purpose – so the same search shortcodes can be used for even hundreds of blogs. If you want to have different outputs on one site and on the other, then you can create two separate search bars (or duplicate one, and change the settings after), and place one on the first and the other on the second.