Reproduction steps:
1. Open the attached “ProfilingIssue.zip” project
2. Open the Profiler Window (Window > Analysis > Profiler)
3. Build and Run for WebGL
4. Interact with the Player in the Browser
5. Observe that the Profiler Window has connected to the Player
6. Click on the Profiler Window

Expected result: The Profiler target is the Player and “Autoconnected Player” text is visible
Actual result: The Profiler target changes to “Play mode” or “Edit mode”

Reproducible with: 2021.3.35f1, 2022.3.20f1, 2023.2.12f1, 2023.3.0b9

Reproducible on: Windows 11
Not reproducible on: no other environment tested

Note:
- Only reproduces in URP
- Also reproduces with Memory Profiler