The "other system' I was referring to was the LIRR.
Neither LIRR nor NJT "stores" train in Penn Station platforms. NJT may relay for a good half hour, still not worth their while to head to Sunnyside to loop back, but LIRR mostly flows through to West Side Yard, inbound generally on track 20 & 21 from their exclusive East River line 3, which do not even lead to the Hudson tunnels, or they had back east in 10 minutes or so if they come in on other tracks.
Nothing taller than 12'10" will fit into LIRR East Side Access, which is 40% of their service, and their trains roam all over their branches and city terminals every day, so there will never be any such thing as universal equipment. There is nothing to agree to.
For the tiny minority of people who are travelling between LIRR & NJT, you don't up end the operations, reliability, and equipment rosters of both systems to accommodate. And even then, with 10 LIRR branches and 7 NJT lines, 70 permutations, the odds of satisfying anyone’s need with route pairings ranges from infinitesimal to none. Transferring and walking from track 4 to track 18 can be done in 2 minutes. For anyone headed from New Jersey to western Queens, there's also the subway for $2.90.
The NJT-equipped trains between New Haven and Secaucus for those special events had about 300 people on board each of the 10 car trains east of Manhattan. They have long since been discontinued as a marketing failure. OTOH, the thru-ticketing arrangement between LIRR and the Meadowlands was used by thousands.
