Implement $project pipeline aggregation stage

[w84thesun]

Description

Closes #2382.

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

[codecov]

Codecov Report

Merging #2383 (d8e32b4) into main (b1cc4e8) will decrease coverage by 38.25%.
The diff coverage is 0.00%.

Additional details and impacted files

@@             Coverage Diff             @@
##             main    #2383       +/-   ##
===========================================
- Coverage   64.65%   26.41%   -38.25%     
===========================================
  Files         397      398        +1     
  Lines       19643    19681       +38     
===========================================
- Hits        12701     5199     -7502     
- Misses       6029    13925     +7896     
+ Partials      913      557      -356     

Impacted Files	Coverage Δ
...nal/handlers/common/aggregations/stages/project.go	0.00% <0.00%> (ø)
...rnal/handlers/common/aggregations/stages/stages.go	0.00% <ø> (-20.00%)	⬇️
internal/handlers/common/projection.go	0.00% <0.00%> (-73.86%)	⬇️
internal/handlers/common/projection_iterator.go	0.00% <0.00%> (-90.00%)	⬇️
internal/handlers/commonerrors/error.go	0.00% <ø> (-48.39%)	⬇️
internal/handlers/commonerrors/errorcode_string.go	80.00% <ø> (ø)

... and 123 files with indirect coverage changes

Flag	Coverage Δ
integration	5.04% <0.00%> (-52.79%)	⬇️
mongodb	5.04% <0.00%> (-0.01%)	⬇️
pg	?
unit	25.04% <0.00%> (-0.21%)	⬇️

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

[chilagrow]

Looks great, just tiny comments

[noisersup]

Apart from comment above looks good to me!

[rumyantseva]

LGTM