: Putting music theme A onto theme B background A friend has shown me 2 piano books she has where a classical piece is used as the background theme and another non-classical piece is overlaid at intervals. For example, a piece

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A technique I liked to practice in my compositions without ever asking for an expression for it.
Now I’ve found the term “blend”:
In the following video a comment says: the tunes have been blended into the music.

Searching for blend two songs I came to the term you mention: mash-up
and this here: www.wikihow.com/Blend-Two-Songs-Together Well, I think merging, mixing, mash-up are all terms derived from the music technic scene:

A mash-up is similar to a DJ mix in that the DJ provides few, if any, significant new sounds to the music, other than those required to blend already-existing music. A mash-up differs from a DJ mix in that two or more songs are playing simultaneously, rather than one after the other. In general, in a mash-up, each original source will contribute more-or-less equally to the final work. There is no requirement, however, that the entire mix of each source be used; a common mash-up technique is to use the vocals from one song mixed with the music of another. Songs involved in a mash-up can be indicated with Mash-up Relationship Type.
musicbrainz.org/doc/Terminology But the idea to mix two or more themes together in a composition is much older: already in Baroque era there was a practice to sing several songs together like canons or a quodlibet.
In my view this technic could be simply called a counterpoint (using another existing song as c.p.
One of my favorite examples: Let the sunshine in and Manchester England from the musical HAIR.
P.S.
Another idea: How about assembling? (related with Ensemble for a group of musicians playing together ...
Edit:
When listening today the Asian sketches by Borodin I reminded your question:

He did exactly what you’re asking and “merged” two tunes together . He calls it(double) counterpoint.

The composer told a very surprised and impressed friend about the excellent processing of the themes: “Before I composed this piece, I had already thought up the two themes - the Russian and the Oriental, which were then both brought together in double counterpoint; and only later did I develop the entire accessory for this orchestra picture."
en.wikipedia.org/wiki/In_the_Steppes_of_Central_Asia