This is certainly fixable, although not a trivial thing to do.
Option one is trying to use the same holes. You can use plastic concrete anchors that are used to drive screws into concrete. You'll have to drive the anchor into the hole and cut off the remaining part. Then you can drive a wood screw into there and that will secure the hinge. I tried this once. I guess you'll have much better results if you apply glue to the anchor surface and let it dry before proceeding so that the anchor is better held in the panel. The key is you use an expandable anchor - one that expands when a screw is being driven in.
Now suppose that it doesn't work because the particle panel has incurred much damage and you can't repeat the original assembly. So the solution is to attach the hinges in slightly different way. Here's a drawing for a single hinge (description follows)
Here 1 is the inner surface of the cabinet, 7 is your hinge - the hinge will go unchanged.
2 is a steel plate - you can get those in a local hardware store, they are typically used to connect the drywall framing parts together, they are typically 2-3 millimeters thick and can have a variety of sizes, you need something like 50 by 50 millimeters or larger so that it fully covers the damaged area and you can drive wood screws 3 and 6 into the cabinet panel to reliably secure the plate. I only show two screws here, but you'd better drive four - one near each corner of the plate - or more.
Then come 4 and 5. Each of them is a plain old threaded screw or bolt in pair with a matching nut. You'll have to pass them through the holes in the hinge that were previously used to attach the hinge (to now wrecked panel) and through holes drilled in the plate and drive a nut from the cabinet surface side. The nuts will likely fit into the holes of the panel, otherwise you'll have to cut some for them.
If the door now closes - great, that was the easy scenario. However you may find that because the plate has some width the door will no longer close. If adjusting the adjustment screws on the hinge (of both doors) doesn't help fix that you'll have to mortise the plate into the panel to compensate for the plate thickness which is not an easy thing to do but can be done using a router (most recommended), a chisel (less recommended, still doable), or a Dremel-like tool (not really recommended).