Flag mashup generator

JavaScript (Node.js),  160  159 bytes

I/O format: matrix of 24-bit integers.

Takes input as (a)(b).

a=>F=b=>(g=a=>[...new Set(a.flat())].sort(_=>Math.random()-.5))(a).every((c,i)=>c^=F[c]=B[i%B.length],B=g(b).slice(-g(a).length))?a.map(r=>r.map(c=>F[c])):F(b)

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Commented

Helper function

The helper function \$g\$ takes a matrix as input and returns a shuffled list of the distinct values it holds.

g = a =>
  [...new Set(a.flat())]
  .sort(_ => Math.random() - .5)

Main code

a => F = b =>                 // a[] = first image, b[] = second image
  g(a).every((c, i) =>        // for each color c at position i in the list of shuffled
                              // colors of the first image:
    c ^=                      //   make sure that c is different from the
      F[c] = B[i % B.length], //     substitute color F[c] picked from the second image
    B =                       //   define B[] as:
      g(b)                    //     the list of shuffled colors from the second image
      .slice(-g(a).length)    //     resized to the number of colors in the first image
  ) ?                         // end of every(); if truthy:
    a.map(r =>                //   return a[] with each original color
      r.map(c => F[c])        //   replaced with its substitute color
    )                         //
  :                           // else:
    F(b)                      //   try again

Python 2, 196 bytes

Takes input as two lists of lists, where different elements represent different colors. Anything can be used to represent color: number, hex, string, etc.

from random import*
P=input()
i,j=(list(set(sum(x,[])))for x in P)
d={0:0}
while any(i==d[i]for i in d):y=[];exec'y+=sample(j,len(j));'*len(i);d=dict(zip(i,y))
print[[d[c]for c in r]for r in P[0]]

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---

Explanation:

# get unique elements from input
P=input()
i,j=(list(set(sum(x,[])))for x in P)  
# create dictionary, where colors of first image are keys and colors of second image are values
d={0:0}
# keep creating while no color is paired with same color
while any(i==d[i]for i in d):
  y=[]
  # sum different samples of substitution colors to keep random
  # and to make sure none of the colors are used more than any other color by more than one
  # *len(i) is to ensure that we have enough values to match
  exec'y+=sample(j,len(j));'*len(i)
  d=dict(zip(i,y))
# substitute colors and output result
print[[d[c]for c in r]for r in P[0]]

Jelly, 40 25 bytes

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A monadic link taking a list of two flags as integer matrices and returning the altered version of the first flag. At the moment this assumes there are indexed colours representing each pixel. I've also assumed that each flag can have arbitrary colours.