NIM-082098 acknowledges that a hyperlink target in a web map pop up cannot be changed from '_blank'. In fact, users can set the attribute to other values but they are not implemented. This behaviour is inconsistent within ArcGIS Online, for example the Story Map Tour template opens links in the same tab whereas links the map viewer opens in a new one.
Ideally, ArcGIS Online should allow developers to implement this attribute as they see fit. This could enable users to navigate seamlessly between multiple maps and apps in the same browser window, resulting in an improved user experience.
Otherwise, it would be helpful to update the documentation on supported HTML to reflect the current behaviour: