Hi Marek,
To display the Manager in People search results, we need to make sure that the Manager property has been indexed in search service application.
Form your description, it seems that you have already created a manager property called Manager for the Manager property and mapped the crawled property People:Manager to the managed property Manager.
Then we need to enable Alias and Indexed in Search Settings in User Profile Service Application for Manager property.
After that, please do a full crawl and then search with Mamager:domain\username to see if the corresponding users will be displayed in People search results.
At last, we need to modify the Item_People.html which is the display template for people search results.
In addition to adding the mapping: Manager:Manager, we need to add the code below to display the Manager value in the search results:
<div>
Manager:_#= ctx.CurrentItem.Manager =#_
</div>
More reference:
http://www.ableblue.com/blog/archive/2013/06/05/introduction-to-sharepoint-2013-display-templates/
Best regards,
Victoria