I think that FindMinimum does not accept symbolic matrix variables. Also, DistanceMatrix must be given an explicit DistanceFunction when the data are mixed numerical-symbolic, otherwise we get unexpected results:

DistanceMatrix[{{x}}, {{0}}]

{{1}}

This seems to give an answer:

Xrs1 = RandomReal[{0, 1}, {10, 3}];
Xrs2 = RandomReal[{0, 1}, {40, 3}];
Xdr2 = RandomReal[{-5, 5}, {40, 2}];

d = DistanceMatrix[Xrs1, Xrs2];

dr[Xdr1_] := 
  DistanceMatrix[Xdr1, Xdr2, 
    DistanceFunction -> EuclideanDistance] /. Abs -> Identity;

f[Xdr1_] := Total[(dr[Xdr1] - d)^2, 2];

FindMinimum[f[Array[Xdr1, {10, 2}]], 
 Flatten[Table[{Xdr1[i, j], 0}, {i, 10}, {j, 2}], 1] // N]

I was forced to append an N because of this further strange behaviour of DistanceMatrix:

In[137]:= DistanceMatrix[{{x[1]}}, {0.}, 
 DistanceFunction -> EuclideanDistance]

Out[137]= {{Abs[0. + x[1.]]}}

whereby x[1] in input becomes x[1.] on output.