Thank you for your suggestions, You-Sheng Yang.

Which kernel/firmware version worked?
-> the default Ubuntu 19.04 kernel and firmware. I never installed a kernel or firmware from external sources and all minor updates that were available worked just fine. What also worked: all other versions of Ubuntu I used on my laptop, at least 18.04 and 18.10 too, most probably I also used 16.04 (not sure).

Which kernel/firmware does not work?
-> The versions at the release of Ubuntu 19.10 (on the .iso images) and also the latest update (5.3.0-19-generic).

An important detail could be, that when I boot Ubuntu 19.10 with the old kernel still installed from 19.04 the same behavior can be seen.

$ uname -a
Linux whakaari 5.3.0-19-generic #20-Ubuntu SMP Fri Oct 18 09:04:39 UTC 2019 x86_64 x86_64 x86_64 GNU/Linux

$ dmesg | grep "iwlwifi\|wlp3s0"
[    4.442485] iwlwifi 0000:03:00.0: Found debug destination: EXTERNAL_DRAM
[    4.442487] iwlwifi 0000:03:00.0: Found debug configuration: 0
[    4.442790] iwlwifi 0000:03:00.0: loaded firmware version 29.1654887522.0 op_mode iwlmvm
[    4.495202] iwlwifi 0000:03:00.0: Detected Intel(R) Dual Band Wireless AC 7265, REV=0x210
[    4.509093] iwlwifi 0000:03:00.0: Applying debug destination EXTERNAL_DRAM
[    4.509828] iwlwifi 0000:03:00.0: Allocated 0x00400000 bytes for firmware monitor.
[    4.520392] iwlwifi 0000:03:00.0: base HW address: 5c:e0:c5:63:e2:a5
[    4.838622] iwlwifi 0000:03:00.0 wlp3s0: renamed from wlan0
[    6.741168] iwlwifi 0000:03:00.0: Applying debug destination EXTERNAL_DRAM
[    6.826404] iwlwifi 0000:03:00.0: Applying debug destination EXTERNAL_DRAM
[    6.829668] iwlwifi 0000:03:00.0: FW already configured (0) - re-configuring
[   65.133013] wlp3s0: authenticate with 9c:4f:cf:a1:91:62
[   65.139402] wlp3s0: send auth to 9c:4f:cf:a1:91:62 (try 1/3)
[   65.159113] wlp3s0: authenticated
[   65.166457] wlp3s0: associate with 9c:4f:cf:a1:91:62 (try 1/3)
[   65.274544] wlp3s0: associate with 9c:4f:cf:a1:91:62 (try 2/3)
[   65.378575] wlp3s0: associate with 9c:4f:cf:a1:91:62 (try 3/3)
[   65.482488] wlp3s0: association with 9c:4f:cf:a1:91:62 timed out
[   66.523712] wlp3s0: authenticate with 9c:4f:cf:a1:91:62
[   66.530411] wlp3s0: send auth to 9c:4f:cf:a1:91:62 (try 1/3)
[   66.634538] wlp3s0: send auth to 9c:4f:cf:a1:91:62 (try 2/3)
[   66.644178] wlp3s0: authenticated
[   66.646503] wlp3s0: associate with 9c:4f:cf:a1:91:62 (try 1/3)
[   66.750461] wlp3s0: associate with 9c:4f:cf:a1:91:62 (try 2/3)
[   66.854575] wlp3s0: associate with 9c:4f:cf:a1:91:62 (try 3/3)
[   66.958583] wlp3s0: association with 9c:4f:cf:a1:91:62 timed out
[   68.401500] wlp3s0: authenticate with 9c:4f:cf:a1:91:62
[   68.406079] wlp3s0: send auth to 9c:4f:cf:a1:91:62 (try 1/3)
[   68.510545] wlp3s0: send auth to 9c:4f:cf:a1:91:62 (try 2/3)
[   68.520607] wlp3s0: authenticated
[   68.522474] wlp3s0: associate with 9c:4f:cf:a1:91:62 (try 1/3)
[   68.626486] wlp3s0: associate with 9c:4f:cf:a1:91:62 (try 2/3)
[   68.730468] wlp3s0: associate with 9c:4f:cf:a1:91:62 (try 3/3)
[   68.834507] wlp3s0: association with 9c:4f:cf:a1:91:62 timed out
[   72.949579] wlp3s0: authenticate with 9c:4f:cf:a1:91:62
[   72.954141] wlp3s0: send auth to 9c:4f:cf:a1:91:62 (try 1/3)
[   72.965169] wlp3s0: authenticated
[   72.966514] wlp3s0: associate with 9c:4f:cf:a1:91:62 (try 1/3)
[   73.070456] wlp3s0: associate with 9c:4f:cf:a1:91:62 (try 2/3)
[   73.174478] wlp3s0: associate with 9c:4f:cf:a1:91:62 (try 3/3)
[   73.278529] wlp3s0: association with 9c:4f:cf:a1:91:62 timed out
[   83.224244] wlp3s0: authenticate with 9c:4f:cf:a1:91:62
[   83.229198] wlp3s0: send auth to 9c:4f:cf:a1:91:62 (try 1/3)
[   83.240334] wlp3s0: authenticated
[   83.242503] wlp3s0: associate with 9c:4f:cf:a1:91:62 (try 1/3)
[   83.346574] wlp3s0: associate with 9c:4f:cf:a1:91:62 (try 2/3)
[   83.450513] wlp3s0: associate with 9c:4f:cf:a1:91:62 (try 3/3)
[   83.554484] wlp3s0: association with 9c:4f:cf:a1:91:62 timed out