Just a note for future reference, the term "Power BI Report Server" refers to the on-premise version. The cloud version is usually called the Power BI Service or sometimes just PowerBI.com (which is confusing I know)
But quoting from this post there are a number of places where this function can be disabled https://www.pbiusergroup.com/communities/community-home/digestviewer/viewthread?MessageKey=1ea6ea79-...
There are multiple places you can control export of data from a Report.
First is from the Admin Portal, where you can control which groups can export data and which can't.
Another is from the Power BI Desktop, go to "File - Options and Settings - Options - Report Settings". There you have the export settings for that Report.
Another is on the Power BI site, for a Report, click on the "gearwheel" Icon on the report list, this again allows you to select the level of export - Summarised Data only, Summarised and Underlying data or None.
Note that this controls the Export option available on the three dots in the header of the Visualisation. Another strategy would be to select "Hide the visual header in the reading view" in Power BI Desktop - Options. This turns off the header completely, so the three dot menu is not available (but then neither is drill up/down etc. )
What these settings do not control is the action on "Right Click" on a visualisation. Some visualisations, for example the Pie Chart, allow "See Records" on right-click. This can expose underlying data if, say, the Chart is using an implicit measure for Average salary, the right click "see records" will show the salary details that go to make up the average. The solution in this case is to use a measure, (Modeling, New Measure) not an implicit measure ( built by dropping down the field name and select "Average").
