Add handler's metrics registration

[AlekSi]

Description

For future cursors metrics.

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 #2895 (3085ae6) into main (82f54cd) will decrease coverage by 0.04%.
The diff coverage is 0.00%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #2895      +/-   ##
==========================================
- Coverage   63.60%   63.57%   -0.04%     
==========================================
  Files         451      451              
  Lines       23555    23569      +14     
==========================================
  Hits        14983    14983              
- Misses       7605     7619      +14     
  Partials      967      967              

Impacted Files	Coverage Δ
internal/clientconn/conn.go	53.72% <ø> (+0.56%)	⬆️
internal/clientconn/listener.go	72.08% <0.00%> (-0.74%)	⬇️
internal/handlers/hana/hana.go	0.00% <0.00%> (ø)
internal/handlers/pg/pg.go	74.64% <0.00%> (-4.46%)	⬇️
internal/handlers/sqlite/sqlite.go	0.00% <0.00%> (ø)
internal/handlers/tigris/tigris.go	0.00% <0.00%> (ø)

Flag	Coverage Δ
integration	56.82% <0.00%> (-0.04%)	⬇️
mongodb	4.19% <0.00%> (-0.01%)	⬇️
pg	56.74% <0.00%> (-0.04%)	⬇️
shard-1	41.93% <0.00%> (-0.03%)	⬇️
shard-2	45.17% <0.00%> (+0.08%)	⬆️
shard-3	40.58% <0.00%> (-0.03%)	⬇️
unit	24.27% <0.00%> (-0.02%)	⬇️

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

[noisersup]

LGTM, one question though