Report actual backend name

[AlekSi]

Description

Store the backend's name in the state and make it dynamic to handle PostgreSQL-compatible systems.
Add it to metrics, log messages, etc.

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 #3570 (a7148c4) into main (52fe847) will decrease coverage by 0.35%.
The diff coverage is 78.78%.

@@            Coverage Diff             @@
##             main    #3570      +/-   ##
==========================================
- Coverage   61.48%   61.14%   -0.35%     
==========================================
  Files         426      426              
  Lines       27200    27200              
==========================================
- Hits        16725    16631      -94     
- Misses       9493     9572      +79     
- Partials      982      997      +15     

Files	Coverage Δ
integration/setup/listener.go	79.38% <100.00%> (ø)
internal/backends/backend.go	92.59% <ø> (-0.27%)	⬇️
internal/backends/decorators/dummy/backend.go	0.00% <ø> (ø)
internal/backends/decorators/oplog/backend.go	0.00% <ø> (ø)
internal/backends/hana/backend.go	12.50% <ø> (+1.38%)	⬆️
internal/backends/postgresql/backend.go	57.31% <ø> (-1.02%)	⬇️
internal/backends/sqlite/backend.go	67.74% <ø> (-1.01%)	⬇️
internal/handlers/sqlite/msg_getlog.go	81.65% <100.00%> (+0.16%)	⬆️
internal/util/state/metrics.go	100.00% <100.00%> (ø)
internal/util/state/state.go	100.00% <100.00%> (ø)
... and 5 more

... and 12 files with indirect coverage changes

Flag	Coverage Δ
filter-true	57.64% <48.27%> (-0.36%)	⬇️
hana-1	2.89% <6.89%> (-0.01%)	⬇️
integration	57.64% <48.27%> (-0.36%)	⬇️
mongodb-1	4.51% <0.00%> (-0.01%)	⬇️
postgresql-1	41.06% <20.68%> (-0.09%)	⬇️
postgresql-2	40.65% <20.68%> (+<0.01%)	⬆️
postgresql-3	39.53% <37.93%> (-0.01%)	⬇️
sort-false	57.64% <48.27%> (-0.36%)	⬇️
sqlite-1	?
sqlite-2	41.18% <24.13%> (+<0.01%)	⬆️
sqlite-3	39.49% <31.03%> (-0.01%)	⬇️
unit	22.38% <61.29%> (-0.03%)	⬇️

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

[rumyantseva]