Android doesn't have a distinction between "text fields" and "text areas" like HTML has. One control, called
EditText, but they're both the same underlying control) handles all text fields, whether they're editable or not, and whether they're one line or several.
That's not to say that every text field behaves the same way.
TextView gives the app the opportunity to set hints for the keyboard, to tell it what kind of text is being edited (called the input type). One input type tells the keyboard that the text being entered is a search filter. The keyboard's free to interpret these hints however it likes: in this case, Google Keyboard doesn't show any word completions/suggestions when the input type is a search filter.
You can avoid this problem by using a different keyboard app which interprets the input type and other hints differently. For example, SwiftKey shows word completions for search filters just like for normal text. (I've just tested it on Google Search.) SwiftKey also supports gesture typing like the default keyboard.
If you're interested, you can learn more about the different hints apps can give to the keyboard by playing with my app IM prove. (It's a free app with no ads, so all I get if you use it is a warm fuzzy feeling when you leave a good rating.) With this app, you can set all the hints, and see what effect they have on your keyboard. You can compare different keyboards this way.