If it was hide glued, I'd recommend sticking (har har) with it.
You can easily test for hide glue, a tiny bit of denatured alchohol will crystalize hide glue. With heat, it will revert back to glue, so it won't ruin the joint. Hot water will also reactivate the glue, so a drop or so and the glue should become tacky again.
You can buy bottled liquid hide glue (Old Brown Glue or Titebond), but check the freshness date, if its more than a year old, don't go with it. You can test it with a drop between your thumb and pointer finger, if it doesn't get stringy/tacky within 30 odd seconds of pulling your fingers apart, its probably gone bad.
Hot hide glue is more involved. If you want to embark upon the joys of hot hide glue, thats a longer discussion. Well worth the effort, IMHO.
Liquid has a longer open time than (unmodified) hot, but hide glue gels. You want the joint as locked in as possible before the gel starts.
To hide glue veneer without ruining the finish, you want to use a dry heat to warm the veneer and substrate.
Hairdryer on low comes to mind.
The old hide glue will reactivate with heat and join with the new glue. One of the many benefits of hide glue.
A little warm water in the joint might also help soften it. Not too much, it weakens the gram strength of the glue. The hot water will soften the old glue.
If its not hide glue, adding hide glue might work. Hide glue is one of the few glues that will adhere to modern glues, but its really hit or miss.
Hide glue won't hurt the finish, but removing it might. I'd try waxing the finished surfaces. That will keep the glue from bonding.
Can you tell that I've been reading "Hide Glue" book from Stephen Shepard? He has a whole chapter on veneer repair.
If there is a "modern" glue, you'd want to try to remove it before adding hide glue. A cabinet scraper and steady hands
Jim