OS X Cmd+G starts searching for same text in different programs
It's a 'feature' - though its use in a Find dialog may have been almost accidental initially, it is actually an intentional usage.

OS X, in common with several other flavours of unix, has two copy buffers.  

The one you know &amp; love, cut/copy/paste itself on  Cmd ⌘   X ,  Cmd ⌘   C  &amp;  Cmd ⌘   V 
&amp; another, known as Kill &amp; Yank, which can be called from  Ctrl ⌃   K  &amp;  Ctrl ⌃   Y 
though incidentally in Stack Exchange's markdown language,  Ctrl ⌃   K  will format as code instead.  
Kill is the equivalent of Cut, rather than Copy, but Yank is equivalent to Paste.

It is actually this kill/yank buffer that most Find/replace dialogs use.
There is no way to avoid that, I'm afraid.