Use the standard *mongo.WriteError type

[AlekSi]

Description

To avoid introducing our own type.

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), Milestone (Next), Labels, Project and project's Sprint fields.
• I marked all done items in this checklist.

[codecov]

Codecov Report

Merging #3705 (9f0c367) into main (a58d9f8) will decrease coverage by 0.10%.
The diff coverage is 69.76%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #3705      +/-   ##
==========================================
- Coverage   76.27%   76.17%   -0.10%     
==========================================
  Files         323      323              
  Lines       23029    23029              
==========================================
- Hits        17566    17543      -23     
- Misses       4422     4444      +22     
- Partials     1041     1042       +1     

Files	Coverage Δ
internal/handlers/sqlite/msg_delete.go	76.23% <100.00%> (ø)
internal/handlers/sqlite/msg_insert.go	79.52% <100.00%> (ø)
internal/handlers/sqlite/msg_update.go	74.13% <100.00%> (ø)
internal/handlers/sqlite/msg_findandmodify.go	66.40% <7.14%> (ø)

... and 12 files with indirect coverage changes

Flag	Coverage Δ
filter-true	72.20% <69.76%> (-0.11%)	⬇️
hana-1	?
integration	72.20% <69.76%> (-0.11%)	⬇️
mongodb-1	5.47% <0.00%> (?)
postgresql-1	52.49% <58.13%> (ø)
postgresql-2	51.64% <32.55%> (ø)
postgresql-3	50.69% <41.86%> (ø)
sort-false	72.20% <69.76%> (-0.11%)	⬇️
sqlite-1	51.67% <58.13%> (-0.16%)	⬇️
sqlite-2	50.86% <32.55%> (+<0.01%)	⬆️
sqlite-3	49.90% <41.86%> (ø)
unit	30.08% <0.00%> (ø)

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