Tests for startSession

[rumyantseva]

Description

Closes #1554.

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

Attention: 10 lines in your changes are missing coverage. Please review.

Comparison is base (9de2931) 76.18% compared to head (20805e2) 75.03%.
Report is 2 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #4112      +/-   ##
==========================================
- Coverage   76.18%   75.03%   -1.16%     
==========================================
  Files         349      349              
  Lines       22157    22172      +15     
==========================================
- Hits        16880    16636     -244     
- Misses       3964     4266     +302     
+ Partials     1313     1270      -43     

Files	Coverage Δ
integration/setup/setup.go	73.75% <0.00%> (-5.64%)	⬇️

... and 21 files with indirect coverage changes

Flag	Coverage Δ
filter-true	67.32% <0.00%> (-0.47%)	⬇️
hana-1	?
integration	67.32% <0.00%> (-0.47%)	⬇️
mongodb-1	4.74% <0.00%> (+0.01%)	⬆️
postgresql-1	49.76% <0.00%> (+1.54%)	⬆️
postgresql-2	47.82% <0.00%> (+0.05%)	⬆️
postgresql-3	48.97% <0.00%> (+1.06%)	⬆️
sqlite-1	48.93% <0.00%> (+1.52%)	⬆️
sqlite-2	46.96% <0.00%> (-0.07%)	⬇️
sqlite-3	48.13% <0.00%> (+1.06%)	⬆️
unit	33.14% <ø> (+0.06%)	⬆️

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