How to reproduce:
1. Open the “Issue IN-72928” project
2. Open the “SampleScene”
3. Add a new GameObject to the Hierarchy
4. In the Inspector, add a Skinned Mesh Renderer component to the newly added GameObject
5. Open the Mesh selection window in the Skinned Mesh Renderer component
6. Double-click on the "Sphere" Mesh
7. Check if the selected “Sphere” Mesh has BlendShapes setting in the Skinned Mesh Renderer component, if not select another “Sphere” Mesh
8. Open the Mesh selection window in the Skinned Mesh Renderer component
9. Select any Mesh
10. Repeat the step 9

Expected result: The GameObject's selected Meshes are changing and visible in the Scene View. The selection also reflected in the Mesh field in the Skinned Mesh Renderer component
Actual result: The GameObject's Mesh does not change, and the selected Mesh is not reflected in the Mesh field in the Skinned Mesh Renderer component

Reproducible in: 2021.3.37f1, 2022.3.24f1, 2023.2.19f1, 6000.0.0b15

Reproduced on: Windows 10
Not reproduced on: No other environments tested

Note: In rare cases, the reproduction might be inconsistent, but after repeating the reproduction steps, the bug starts occurring