Implement projection fields assignment

[w84thesun]

Description

Closes #2446.

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 #2484 (91e232b) into main (f0958a9) will increase coverage by 0.11%.
The diff coverage is 93.33%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #2484      +/-   ##
==========================================
+ Coverage   64.54%   64.65%   +0.11%     
==========================================
  Files         397      397              
  Lines       19631    19643      +12     
==========================================
+ Hits        12670    12701      +31     
+ Misses       6045     6029      -16     
+ Partials      916      913       -3     

Impacted Files	Coverage Δ
internal/handlers/common/projection.go	73.85% <93.33%> (+11.44%)	⬆️

... and 3 files with indirect coverage changes

Flag	Coverage Δ
integration	57.82% <93.33%> (+0.12%)	⬆️
mongodb	5.05% <0.00%> (-0.01%)	⬇️
pg	57.75% <93.33%> (+0.12%)	⬆️
unit	25.24% <0.00%> (-0.10%)	⬇️

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[chilagrow]

Just a single comment while looking at tests 🤗

[noisersup]

👍