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No. Jira won't let you do this for good reason. Can you explain why you want to link the same story to two different Epics?
If you still needed to do this you could clone the issue and associate them in turn to the two Epics. Of course, this would get very silly very quickly the more times you do this.
Danno is right in that you cannot link a Story/Task in Jira to more than one Epic if/when you are using a Jira Epic link.
This is somewhat counterintuitive, as Atlassian defines an Epic this way: "An epic captures a large body of work—performance-related work, for example—in a release. It's essentially a large user story that can be broken down into a number of smaller stories. An epic can span more than one project if multiple projects are included in the board where the epic is created."
You can however do what you describe by using another Jira issue link type and I see customers doing this all the time.