That piece of drain pipe with two hose barbs looks like it was home-made. Don't try to fix or replace it. You can buy something similar:

The above part has the added benefit that water from the sink won't enter the filter hoses.

Or you could use two standard connectors for a single dishwasher. You'll also have to replace the rest of the trap assembly with compatible PVC compression fit parts.

Disconnect the entire assembly between these two marked points:

take it whole to a good hardware or plumbing supply store and try to assemble something using products off the shelf that that does the job and will fit the same way when assembled.

While it is disassembled you should place a sandwich bag with a rubber band over the pipe coming out of the wall to avoid smells coming out of it.

That piece of drain pipe with two hose barbs looks like it was home-made. Don't try to fix or replace it. You can buy something similar:

The above part has the added benefit that water from the sink won't enter the filter hoses.

Or you could use two standard connectors for a single dishwasher. You'll also have to replace the rest of the trap assembly with compatible PVC compression fit parts.

Disconnect the entire assembly between these two marked points:

take it whole to a good hardware or plumbing supply store and try to assemble something using products off the shelf that that does the job and will fit the same way when assembled.

While it is disassembled you should place a sandwich bag with a rubber band over the pipe coming out of the wall to avoid smells coming out of it.

Incidentally and while you are ducked under there, if you move the attachment point between the drain popup and its knob further away from the drain and probably to a lower hole, it should become easier to operate, and might seal better and drain better. Choose a hole that allows full travel and doesn't max out at either end.

That piece of drain pipe with two hose barbs looks like it was home-made. Don't try to fix or replace it.

There are lots of similar ones you can buy that will connect one hose from a dishwasher or filter like yours. I suggest you disconnect the entire assembly between these two marked points:

take it whole to a good hardware or plumbing supply store and try to assemble something using products off the shelf that includes two standard dishwasher connections and also assembles so it will match the two existing end points.

While it is disassembled you should place a sandwich bag with a rubber band over the pipe coming out of the wall to avoid smells coming out of it.

There are also some pieces available that will allow you to connect both hoses. If your store has one, try to make it fit. Some of these join them together and then exit with a similarly small hose. I don't know if that's ok for your filter, you need to find out.

That piece of drain pipe with two hose barbs looks like it was home-made. Don't try to fix or replace it. You can buy something similar:

The above part has the added benefit that water from the sink won't enter the filter hoses.

Or you could use two standard connectors for a single dishwasher. You'll also have to replace the rest of the trap assembly with compatible PVC compression fit parts.

Disconnect the entire assembly between these two marked points:

take it whole to a good hardware or plumbing supply store and try to assemble something using products off the shelf that that does the job and will fit the same way when assembled.

While it is disassembled you should place a sandwich bag with a rubber band over the pipe coming out of the wall to avoid smells coming out of it.