Please check this video as proof. To prove it, I'm changing the 'placeholder' text to 'TEST' before I scroll down the screen. If the menu does not touch the search node, it will not change at all, and that is what should happen normally. But as you can see, it reverts back to 'SZUKAJ…', which means that the whole section is altered (probably replaced with a copy), resulting in loss of every previously attached event handlers – which breaks the search, or any 3rd party element that uses scripts for interaction: https://youtu.be/kRmR8Q3YHcU
This is a very bad practice, and it should not be used if possible – as it results in loss of attached script handlers. Unfortunately there is absolutely no way for the search to prevent this. I would suggest a solution if there was any, but it is not caused by the search itself, but by a script that alters the node structure of the header section.
You might have to ask the menu developer if this is something that can be turned off, or if there is any way for the menu to simply hide the elements, but not to make copies and change the header structure.
If you like my products, don't forget to rate them on codecanyon :)