JIRA Client cannot download the list of all link types from JIRA directly. Instead, it "learns" them from the issues that are already downloaded.
Most of the time, this does not present a problem since, in a production environment, the pool of the issues downloaded upon the very first sync contains all the link types that are in active use. Therefore, JIRA Client "learns" all the links that matter. Everytime JIRA Client encounters an issue with a new link, it adds it to the list.
In a situation, where were JIRA instance is relatively empty, and initial sync does not load many issues and these issues do not contain any links (e.g. test environment), JIRA Client will not encounter any links, thus the links drop down list in the Issue Details window will be empty.
To summarize, sometimes having an empty or incomplete list of links in JIRA Client is an expected behavior. You can "teach" JIRA Client to recognize all the links that exist in your JIRA instance by creating 2 sample issues and linking them to each other using all available link types.