Can I somehow make a search request for related videos (related to video from given id)?
Something similar to vimeo.categories.getRelatedVideos, but for vimeo.videos
The methods you described are part of the old API. It is deprecated and will not continue to be supported. The new api (https://developer.vimeo.com/api) exposes some related videos through https://developer.vimeo.com/api/endpoints/videos#GET/videos/%7Bvideo_id%7D/videos.