To reproduce:
1. Open any project, open Profiler window
2. In CPU usage tab, change preview type to something that's not Timeline
3. Select different Profiler (say Memory)
4. Enter Play mode
5. Observe Profiler window

Expected: Profiler keeps the focus and preview type from Editor mode
Actual: Profiler resets the selection and preview to Cpu Usage - Timeline

Reproduced in: 2018.4.0f1 (partially), 2018.4.6f1 (partially), 2019.1.14f1 (partially), 2019.2.1f1 (partially), 2019.3.0a7 (partially), 2019.3.0a8, 2019.3.0a11
Not reproduced in: 2017.4.0f1, 2017.4.31f1

Note1: the issue had multiple regressions, only marking the most recent one
Note2: even if the CPU Profiler is disabled, entering Play mode still shows the details of CPU Profiler
Note3: this issue also extends to the GPU Profiler Module, and at least on 2019.3.0a10, also to the Audio and Memory modules.