How can I color a cylinder inside a sphere?

sphere = 
  SphericalPlot3D[5, {θ, 0, π}, {φ, 0, 2 π}, 
   RegionFunction -> Function[{x, y, z}, x^2 + y^2 >= 2^2], 
   Boxed -> False, Axes -> False, MeshStyle -> White, 
   ColorFunction -> Function[{x, y, z}, ColorData["Rainbow"][z]], 
   PlotStyle -> Opacity[.5]];
cylinder = 
  ContourPlot3D[
   x^2 + y^2 == 2^2, {x, y, z} ∈ Ball[{0, 0, 0}, 5], 
   RegionBoundaryStyle -> None, ContourStyle -> FaceForm[Yellow, Red],
    Mesh -> None];
Show[sphere, cylinder]

• If we do not use the expression of sphere and cylinder, we could use OpenCascadeLink and FEM to separate the exterior sphere and interior cylinder faces. see the answer by @user21

Clear["Global`*"];
Needs["OpenCascadeLink`"];
Needs["NDSolve`FEM`"];
reg1 = Ball[{0, 0, 0}, 5];
reg2 = Cylinder[{{0, 0, -5}, {0, 0, 5}}, 2];
shape = OpenCascadeShapeDifference[
   OpenCascadeShape /@ {reg1,reg2}];
bmesh = OpenCascadeShapeSurfaceMeshToBoundaryMesh[shape, 
   "ShapeSurfaceMeshOptions" -> {"AngularDeflection" -> .05}];
mesbm = Flatten[
   MeshElementSplitByMarker[
    MeshOrderAlteration[bmesh, 1]["BoundaryElements"]]];
{sphere, cylinder} = 
  MeshRegion[bmesh["Coordinates"], Polygon[ElementIncidents[#]]] & /@ 
   mesbm;
{RegionPlot3D[sphere, 
   ColorFunction -> Function[{x, y, z}, ColorData["Rainbow"][z]], 
   PlotStyle -> Opacity[.5], MeshFunctions -> {#3 &, ArcTan[#1, #2] &},
    Mesh -> 20, MeshStyle -> White, Boxed -> False], 
  Graphics3D[{EdgeForm[], FaceForm[Yellow, Red], cylinder}]} // Show

• Reply to comment for sphere and cone ets.

Clear["Global`*"];
Needs["OpenCascadeLink`"];
Needs["NDSolve`FEM`"];
clip[reg1_, reg2_] := Module[{shape, bmesh, mesbm, regs},
  shape = OpenCascadeShapeDifference[OpenCascadeShape /@ {reg1, reg2}];
  bmesh = 
   OpenCascadeShapeSurfaceMeshToBoundaryMesh[shape, 
    "ShapeSurfaceMeshOptions" -> {"AngularDeflection" -> .05}];
  mesbm = Flatten[MeshElementSplitByMarker[bmesh["BoundaryElements"]]];
  regs = 
   MeshRegion[bmesh["Coordinates"], Polygon[ElementIncidents[#]]] & /@
     mesbm;
  {RegionPlot3D[regs[[1]], 
     ColorFunction -> Function[{x, y, z}, ColorData["Rainbow"][z]], 
     PlotStyle -> Opacity[.5], 
     MeshFunctions -> {#3 &, ArcTan[#1, #2] &}, Mesh -> 20, 
     MeshStyle -> White, Boxed -> False], 
    Graphics3D[{EdgeForm[], FaceForm[Yellow, Red], 
      regs[[Complement[
         bmesh["BoundaryElementMarkerUnion"], {1}]]]}]} // Show]

reg1 = Ball[{0, 0, 0}, 5];
reg2 = Cone[{{1, 3, -5}, {2, -7, 5}}, 4];
clip[reg1, reg2]

reg1 = Ball[{0, 0, 0}, 5];
reg2 = TransformedRegion[Ellipsoid[{0, 0, 0}, {7, 2, 3}], 
   RotationTransform[π/3, {1, 1, 1}]];
clip[reg1, reg2]

reg1 = Ball[{0, 0, 0}, 5];
reg2 = FilledTorus[{3, 2, 1}, {3, 5}];
clip[reg1, reg2]