Improve MongoDB/FerretDB error checking in tests

[AlekSi]

Description

The main error message should come from MongoDB.

Readiness checklist

• I added/updated unit tests (and they pass).
• I added/updated integration/compatibility tests (and they pass).
• 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 #3143 (b613a2d) into main (9673b2d) will decrease coverage by 0.20%.
The diff coverage is 85.71%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #3143      +/-   ##
==========================================
- Coverage   75.47%   75.27%   -0.20%     
==========================================
  Files         393      393              
  Lines       21868    21872       +4     
==========================================
- Hits        16504    16464      -40     
- Misses       4401     4448      +47     
+ Partials      963      960       -3     

Files Changed	Coverage Δ
integration/helpers.go	86.87% <80.00%> (+0.09%)	⬆️
integration/setup/test_helpers.go	100.00% <100.00%> (ø)

... and 18 files with indirect coverage changes

Flag	Coverage Δ
hana	?
integration	71.71% <85.71%> (-0.20%)	⬇️
mongodb	5.38% <57.14%> (-0.02%)	⬇️
pg	65.24% <71.42%> (-0.01%)	⬇️
shard-1	54.89% <78.57%> (-0.36%)	⬇️
shard-2	53.13% <57.14%> (+0.06%)	⬆️
shard-3	58.04% <71.42%> (+0.69%)	⬆️
sqlite	37.97% <50.00%> (+3.28%)	⬆️
unit	24.65% <ø> (ø)

Flags with carried forward coverage won't be shown. Click here to find out more.

[chilagrow]

🚀

[rumyantseva]