Is it a paper pass, since you mention it has a start date?

The validity period of a paper pass cannot be changed. A pass can normally be exchanged or be refunded (with costs). If the pass was bought on interrail.eu, see:

https://www.interrail.eu/en/interrail-passes/ordering-info/refund-exchange-policy

If the pass was bought elsewhere, then it needs to be exchanged or refunded there.

I’ve got a Mobile Pass which was gifted to me.

A mobile pass doesn't have a start date until you choose it in the 3rd activation step in the app. Is 5 June maybe the latest possible start date?

In any case, a mobile pass can normally be exchanged, as you can see on the page I mentioned.

I'm not sure how that works when someone else gave you a pass, you it's probably best to ask customer service first:

https://eurail.zendesk.com/hc/en-001/requests/new

Yes, indeed. My latest possible start date is 5th of June. Is there any way i can postpone this date?

The normal way would be to do a pass exchange.

If you have any special reason to postpone, you could ask customer service.