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installing over an existing installation
If you did install over an existing installations you may be able to fix this, with either of the following 2 methods.
1) Menu: Tools > Configure build lazarus
Make sure to check "Clean all" and then hit build.
For an unknown reason this solved the issue and the 32 bit version of Lazarus on the 32 bit Windows could now compile everything. So thank you kindly, good Sir 
If you may allow me the question: What part in the installation, did I screw up? I kinda want to learn from mistakes.
I can not say for 100% sure how you got into the error. Especially if you installed into a new folder.
But a bit of background why it happens.
Each pascal file (SomeUnit.pas) will be compiled into SomeUnit.ppu and SomeUnit.o
The 2 resulting files, are only readable to the exact same version of the compiler.
- Fpc 3.2 can not read fpc 3.0.4 or 3.3.1 ppu files, and vice versa
- 32 bit fpc can not read 64 bit fpc ppu files and vice versa.
So when you upgrade, and an old ppu remains in the folder (because maybe the new replacement ppu goes into a different subfolder), then that one old ppu will cause issues.
Rebuilding the IDE with "clean all" usually (rules are bit more complex) finds any ppu file (including such old ones) and removes them.
The other cause for issues like this, is if users edit the "path" settings in the project options (or in package options).
But I do not think you did that, because the "Compile IDE fix" does not help for this.
Sometimes users include units from a package by adding the path. That can lead to ppu files being created twice, in the package and in the project dir. Then only one gets updated, and the outdated one causes problems. (even if the compiler is the same)
The IDE will warn in most cases, if a user attempts to add such paths. But it might not catch every case.
In conclusion:
One of your ppu files was broken (old, outdated, wrong sub folder, or simply corrupted (disk error, antivirus, compiler terminated while writing it)). The exact reason is hard to determinate.
Rebuilding the IDE, did fix that.
Note in some cases you might have to remove the ppu using the search in the explorer.
Missing ppu will be rebuild. (unless they have copies, but in other folders, but if you delete those too, then they will be rebuild)
Only if you have a closed source 3rd party package (that you bought), then this may not ship sources but ppu only => such ppu can not be rebuild (missing source).