Java 8u71 failed to install with Error 1610

We have received multiple incident reports indicating Java installation failure with exception error message "KnownProductCodeInstalledJavaTracker". This issue is reported with Windows 7 as well as Windows 10.
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I just have the same problem, but on Windows 7 itself! I already worked on this problem last summer with 8u60, but didn't finish it at that time. Now, again, JRE had to be used, so I tried to install it with latest 8u65/66 last week. Still no success. After that, I found this thread but no useful information at that time. So I installed the last working version 8u51 I had luckily lying around on my hard drive and then copied the files of 8u65 zip download over them and got it to work. Today I tried the latest 8u71, still no luck. I have the same/similar error as mentioned by @3067688 in November 2015.

The jusched.log file says:
[2016/01/11 17:09:41.664, jre-8u66-windows-i586.exe (PID: 5220, TID: 4692), JavaEnvironment.cpp:0 (JavaEnvironment::detect)]
TRACE: Exiting JavaEnvironment::detect (entered at JavaEnvironment.cpp:472)
[2016/01/11 17:09:41.666, jre-8u66-windows-i586.exe (PID: 5220, TID: 4692), wrapper.cpp:507 (WinMain)]
ERROR: Exception with message 'KnownProductCodeInstalledJavaTracker.cpp(205) at KnownProductCodeInstalledJavaTracker::next(): MsiEnumProducts(97) failed with error=[1610]' caught
[2016/01/11 17:09:41.669, jre-8u66-windows-i586.exe (PID: 5220, TID: 4692), wrapper.cpp:509 (WinMain)]
INFO: Wrapper exit code: 1
and from today:
[2016/01/21 16:55:55.432, jre-8u71-windows-i586.exe (PID: 2140, TID: 4100), JavaEnvironment.cpp:391 (`anonymous-namespace'::adjustVersion)]
INFO: Detected version mapping from 1.8.0_51 to 1.8.0_66 in {26A24AE4-039D-4CA4-87B4-2F83218051F0} Java installation.
[2016/01/21 16:55:55.435, jre-8u71-windows-i586.exe (PID: 2140, TID: 4100), JavaEnvironment.cpp:327 (`anonymous-namespace'::detectJreMode)]
TRACE: Detect mode of {26A24AE4-039D-4CA4-87B4-2F83218051F0} Java installation
[2016/01/21 16:55:55.438, jre-8u71-windows-i586.exe (PID: 2140, TID: 4100), JavaEnvironment.cpp:451 (`anonymous-namespace'::getInstalledJava)]
TRACE: InstalledJava(1.8.0_66|1.8.0_66|JRE|32bit|D:\Java\jre8|{26A24AE4-039D-4CA4-87B4-2F83218051F0}|10)
[2016/01/21 16:55:55.454, jre-8u71-windows-i586.exe (PID: 2140, TID: 4100), JavaEnvironment.cpp:0 (JavaEnvironment::detect)]
TRACE: Exiting JavaEnvironment::detect (entered at JavaEnvironment.cpp:472)
[2016/01/21 16:55:55.457, jre-8u71-windows-i586.exe (PID: 2140, TID: 4100), wrapper.cpp:507 (WinMain)]
ERROR: Exception with message 'KnownProductCodeInstalledJavaTracker.cpp(205) at KnownProductCodeInstalledJavaTracker::next(): MsiEnumProducts(97) failed with error=[1610]' caught
[2016/01/21 16:55:55.460, jre-8u71-windows-i586.exe (PID: 2140, TID: 4100), wrapper.cpp:509 (WinMain)]
INFO: Wrapper exit code: 1
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