Add basic postcondition checker for errors

[AlekSi]

Description

Closes #1095.

Readiness checklist

• I added/updated unit tests.
• I added/updated integration/compatibility tests.
• I added/updated comments and checked rendering.
• I made spot refactorings.
• I updated user documentation.
• I ran task all, and it passed.
• I ensured that PR title is good enough for the changelog.
• (for maintainers only) I set Reviewers (@FerretDB/core), Labels, Project and project's Sprint fields.
• I marked all done items in this checklist.

[codecov]

Codecov Report

Merging #2607 (ffa42e6) into main (0665e99) will decrease coverage by 38.15%.
The diff coverage is 78.57%.

Additional details and impacted files

@@             Coverage Diff             @@
##             main    #2607       +/-   ##
===========================================
- Coverage   65.93%   27.78%   -38.15%     
===========================================
  Files         416      417        +1     
  Lines       20512    20533       +21     
===========================================
- Hits        13524     5705     -7819     
- Misses       6035    14242     +8207     
+ Partials      953      586      -367     

Impacted Files	Coverage Δ
internal/handlers/pg/pgdb/databases.go	23.71% <70.00%> (-48.63%)	⬇️
internal/util/contract/contract.go	83.33% <83.33%> (ø)

... and 137 files with indirect coverage changes

Flag	Coverage Δ
integration	5.39% <0.00%> (-53.48%)	⬇️
mongodb	5.39% <0.00%> (-0.01%)	⬇️
pg	?
unit	26.21% <78.57%> (+0.05%)	⬆️

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[w84thesun]

👍