I've noticed this mysteriously amber glue as well in a few old
pianos -- fortunately only on wood joints that didn't need to come
apart -- and I had assumed it was hide glue with some sort of
additive. Perhaps it is casein glue, which is sold as a powder and
mixed with water; no glue pot is needed.

I note also in a reference book that casein glue can be used for gluing
dissimilar materials, such as paper labels to glass and metal. There
is a certain chemical similarity to milk paint, which is notoriously
difficult to remove from old woodwork.

By World War One, vegetable glues based on cassava root (and other
plants) were being used, mostly for veneering panels. So that's another
possibility.

The guys at Kranich & Bach were quite inventive, as I know from the
frustrations of working on their straight uprights.  So, I would say
anything is possible as long as the glue was available at the time.

Richard Friedman
Upstate New York