Remove common handlers.Interface

[AlekSi]

Description

Closes #3662.

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 #3753 (7f8654b) into main (3d03b51) will increase coverage by 0.01%.
The diff coverage is 93.93%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #3753      +/-   ##
==========================================
+ Coverage   76.26%   76.28%   +0.01%     
==========================================
  Files         325      324       -1     
  Lines       23112    23116       +4     
==========================================
+ Hits        17627    17633       +6     
- Misses       4441     4444       +3     
+ Partials     1044     1039       -5     

Files	Coverage Δ
internal/clientconn/conn.go	55.49% <100.00%> (ø)
internal/clientconn/listener.go	75.35% <ø> (ø)
internal/handlers/registry/postgresql.go	77.77% <100.00%> (ø)
internal/handlers/registry/registry.go	43.47% <100.00%> (ø)
internal/handlers/registry/sqlite.go	77.77% <100.00%> (ø)
internal/handlers/sqlite/cmd_query.go	32.00% <ø> (ø)
internal/handlers/sqlite/msg_aggregate.go	83.78% <ø> (ø)
internal/handlers/sqlite/msg_buildinfo.go	100.00% <ø> (ø)
internal/handlers/sqlite/msg_collmod.go	0.00% <ø> (ø)
internal/handlers/sqlite/msg_collstats.go	73.68% <ø> (ø)
... and 43 more

... and 8 files with indirect coverage changes

Flag	Coverage Δ
filter-true	72.44% <93.93%> (+0.03%)	⬆️
hana-1	?
integration	72.44% <93.93%> (+0.03%)	⬆️
mongodb-1	5.47% <18.18%> (+0.02%)	⬆️
postgresql-1	51.78% <30.30%> (+0.02%)	⬆️
postgresql-2	51.39% <90.90%> (+<0.01%)	⬆️
postgresql-3	52.56% <30.30%> (+0.14%)	⬆️
sqlite-1	50.92% <30.30%> (+0.05%)	⬆️
sqlite-2	50.55% <90.90%> (+<0.01%)	⬆️
sqlite-3	51.62% <30.30%> (?)
unit	30.13% <27.27%> (+0.02%)	⬆️

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[rumyantseva]

LGTM!