Material used on edges of mass produced veneered furniture  

More veneer?
More veneer?

Veneer tape.
You can buy it in many different woods, varying widths, and a lot of it has a hot glue backing so you can just apply it with a hot iron (no steam).
They are all made slightly wider than the most popular thickness boards so that you don't have to be absolutely precise in applying. The excess sands off pretty easily because it's thin.

Not veneer :/
So I should have been clearer. The edging is not veneered. Many old lane pieces etc were finished on the edges with a fairly opaque substance (not paint) to cover up the sandwiched layers of veneer and substrate. I am restoring numerous pieces and can't identify this stuff... I've queried many. A million thanks to anyone who can help out.

Probably a pigmented varnish.
Make it yourself with oil-based varnish of your choice adding artist's oil paints from those little tubes. Experiment with different color combinations like adding a little white to increase opacity. You might be surprised.

Pigmented varnish?
Pigmented varnish w lacquer finish? You think so? What about pigmented toner (lacquer)?

Sure, then. A compatible pigmented glaze of your choice.
I think you can get away with mixing small amounts of artist's oils with lacquer. I've never tried it. You might need UTCs or dry pigments.