Refactor database implementation to use PostgreSQL instead of MongoDB (#289)

Fixes #228

## Motivation and Context

See #19

## How Has This Been Tested?

- Run locally
- Run in docker compose
- Run infra in minikube
- Unit and integration tests all passing :)

## Breaking Changes

Yes: users will not be able to access any data they used to have in
custom mongodb. We are accepting this breaking change, given the
registry is in experimental development status.

## Types of changes
<!-- What types of changes does your code introduce? Put an `x` in all
the boxes that apply: -->
- [ ] Bug fix (non-breaking change which fixes an issue)
- [ ] New feature (non-breaking change which adds functionality)
- [x] Breaking change (fix or feature that would cause existing
functionality to change)
- [ ] Documentation update

## Checklist
<!-- Go over all the following points, and put an `x` in all the boxes
that apply. -->
- [x] I have read the [MCP
Documentation](https://modelcontextprotocol.io)
- [x] My code follows the repository's style guidelines
- [x] New and existing tests pass locally
- [x] I have added appropriate error handling
- [x] I have added or updated documentation as needed

## Additional context

I have not implemented connection pooling support for simplicity, and
because cloudnative-pg already provides built-in connection pooling with
pgBouncer so if we want it later we can switch to this endpoint easily.

go.mod

@@ -7,10 +7,10 @@ require (
 	github.com/danielgtaylor/huma/v2 v2.34.1
 	github.com/golang-jwt/jwt/v5 v5.3.0
 	github.com/google/uuid v1.6.0
+	github.com/jackc/pgx/v5 v5.7.5
 	github.com/prometheus/client_golang v1.23.0
 	github.com/santhosh-tekuri/jsonschema/v5 v5.3.1
 	github.com/stretchr/testify v1.10.0
-	go.mongodb.org/mongo-driver v1.17.3
 	go.opentelemetry.io/contrib/instrumentation/runtime v0.62.0
 	go.opentelemetry.io/otel v1.37.0
 	go.opentelemetry.io/otel/exporters/prometheus v0.59.1
@@ -25,10 +25,9 @@ require (
 	github.com/davecgh/go-spew v1.1.1 // indirect
 	github.com/go-logr/logr v1.4.3 // indirect
 	github.com/go-logr/stdr v1.2.2 // indirect
-	github.com/golang/snappy v0.0.4 // indirect
 	github.com/grafana/regexp v0.0.0-20240518133315-a468a5bfb3bc // indirect
-	github.com/klauspost/compress v1.18.0 // indirect
-	github.com/montanaflynn/stats v0.7.1 // indirect
+	github.com/jackc/pgpassfile v1.0.0 // indirect
+	github.com/jackc/pgservicefile v0.0.0-20240606120523-5a60cdf6a761 // indirect
 	github.com/munnerz/goautoneg v0.0.0-20191010083416-a7dc8b61c822 // indirect
 	github.com/pmezard/go-difflib v1.0.0 // indirect
 	github.com/prometheus/client_model v0.6.2 // indirect
@@ -36,14 +35,9 @@ require (
 	github.com/prometheus/otlptranslator v0.0.0-20250717125610-8549f4ab4f8f // indirect
 	github.com/prometheus/procfs v0.17.0 // indirect
 	github.com/stretchr/objx v0.5.2 // indirect
-	github.com/xdg-go/pbkdf2 v1.0.0 // indirect
-	github.com/xdg-go/scram v1.1.2 // indirect
-	github.com/xdg-go/stringprep v1.0.4 // indirect
-	github.com/youmark/pkcs8 v0.0.0-20240726163527-a2c0da244d78 // indirect
 	go.opentelemetry.io/auto/sdk v1.1.0 // indirect
 	go.opentelemetry.io/otel/trace v1.37.0 // indirect
 	golang.org/x/crypto v0.41.0 // indirect
-	golang.org/x/sync v0.16.0 // indirect
 	golang.org/x/sys v0.35.0 // indirect
 	golang.org/x/text v0.28.0 // indirect
 	google.golang.org/protobuf v1.36.6 // indirect