Check authorization

[chilagrow]

Description

Closes FerretDB/engineering#141.

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]

❌ 300 Tests Failed:

Tests completed	Failed	Passed	Skipped
1620	300	1320	684

View the top 3 failed test(s) by shortest run time

github.com/FerretDB/FerretDB/v2/integration/query::TestFindAndModifyCommandErrors/SetImmutableID

Stack Traces | 0s run time

No test result found

github.com/FerretDB/FerretDB/v2/integration/query::TestFindAndModifyCompatRemove

Stack Traces | 0s run time

No test result found

github.com/FerretDB/FerretDB/v2/integration/query::TestFindAndModifyCompatUpdateCurrentDate/UnknownType

Stack Traces | 0s run time

No test result found

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[mergify]

@chilagrow this pull request has merge conflicts.