classic shell does nothing to enable/disable taskbar search. It replaces the start menu, and can have a search-box similar to that of windows 7 (in the windows 7 style) or windows XP(in the classic/classic with 2 column styles) This box can be selected by default when the windows key is pressed, so its effectively the same thing. (but you wont get a searchbox on the taskbar)
In addition classic shell can use 3rd party search programs like
https://www.voidtools.com/ quite nicely (but with some settup)