We have many sites all on separate domains but all share some common components. One component is a "Search Box" at the top of each home page that allows you to search for holidays.
The search box is a <form> with a bunch of <input>, <select> <label> and <ul> tags. The labels, dropdowns, inputs all have the same default values across each site.
Question: will the search box been seen as duplicate content when indexed by Google?
All websites have some elements of their page template that are repeated on many pages. That is not something to worry about. Google does not view small amounts of template text and markup as problematic in any way.
Many sites use the same same themes as other sites. Many sites have search boxes. There is just no way that the duplication your speak of is going to cause problems.
Even if you duplicate whole pages within your site, Google will generally just pick one of the duplicates to index and ignore the rest. Google won't penalize your whole site because of it. See What is duplicate content and how can I avoid being penalized for it on my site?
