A common misconception with the SharePoint Managed Metadata Service, where taxonomies are deployed, is that the "Other Labels" or synonyms that you add to the SharePoint Term Store will have an impact on SharePoint search.
In reality, the "Other Labels" in the SharePoint term store are only useful when users are tagging content with managed metadata terms. What this means is that if "HR" is an other label for "Human Resources" in the term store, then if a user begins to type "HR" in the type-ahead tagging capability, SharePoint will direct the user that the correct tag is "Human Resources". This is helpful and makes sure that users are tagging to the preferred form of the concept that has been defined in the term store. It reduces frustration when tagging by pointing synonyms to the correct place in the taxonomy.
However, it does not help with search. Don't worry! there is a way to use synonyms to improve search.
The value of synonyms in SharePoint search is in query expansion. That is, if "HR" and "Human Resources" are synonyms, a user searching for either term will automatically have the query expanded to search for both variants. This is an effective way to help users get complete search result sets. It's an important detail to address to ensure you are delivering the best search experience possible for your users.
Microsoft Technet provides this article with detailed instructions on how to create and maintain a thesaurus in SharePoint Server 2013. There is a Technet blog post with similar content in case you want an article with some more images. The content is relevant for SharePoint Server 2016 as well.
Unfortunately, you cannot import a thesaurus in SharePoint Online.
Taxonomies downloaded for SharePoint from the WAND Taxonomy Library Portal come with a file listing the synonyms for the taxonomy. This file can be used as a thesaurus file for the SharePoint 2013 or SharePoint 2016 search engine. Remember, however, that only a single thesaurus file can be loaded, so if you download multiple taxonomies from WAND, you will need to combine the supplementary synonym files into one to create a master thesaurus file.
This blog post is relevant for SharePoint 2013 and SharePoint 2016