To change the logo of your dokuwiki shown in the title bar of the browser, also used for bookmarks, you have to create a 48×48 .ico logo and overwrite

lib/tpl/dokuwiki/images/favicon.ico

(favicon.icon is the standard name for the FAVorite ICOn https://en.wikipedia.org/wiki/Favicon)

Important notes

To have a well proportioned logo, you have to resize it to a 48px, width or height (whichever is wider), then start a new page 48×48 pixels and paste it inside. Export as favicon.ico.

You can do it with Gimp starting from your big logo.