feat(dbmanager): report pool max size and total connections opened

Add MaxOpenConnections to provider and connection stats, exposed as the
max state of dbmanager_connection_pool_size. Count physical dials through a
counting driver connector and export dbmanager_connections_opened_total;
existing_db pools derive an approximate count from sql.DBStats.

pkg/dbmanager/providers/counting.go

@@ -0,0 +1,51 @@
+package providers
+
+import (
+	"context"
+	"database/sql"
+	"database/sql/driver"
+	"sync/atomic"
+)
+
+// countingConnector wraps a driver.Connector and counts every physical
+// connection it successfully dials. sql.DBStats has no such field, so this is
+// the only way to report the total number of connections ever opened.
+type countingConnector struct {
+	driver.Connector
+	opened *atomic.Int64
+}
+
+func (c *countingConnector) Connect(ctx context.Context) (driver.Conn, error) {
+	conn, err := c.Connector.Connect(ctx)
+	if err == nil {
+		c.opened.Add(1)
+	}
+	return conn, err
+}
+
+// dsnConnector adapts a plain driver.Driver to driver.Connector.
+type dsnConnector struct {
+	dsn string
+	drv driver.Driver
+}
+
+func (c dsnConnector) Connect(context.Context) (driver.Conn, error) { return c.drv.Open(c.dsn) }
+func (c dsnConnector) Driver() driver.Driver                        { return c.drv }
+
+// openCounted is sql.Open with every dialled connection counted in opened.
+func openCounted(driverName, dsn string, opened *atomic.Int64) (*sql.DB, error) {
+	probe, err := sql.Open(driverName, dsn)
+	if err != nil {
+		return nil, err
+	}
+	drv := probe.Driver()
+	probe.Close() //nolint:gosec // G104: probe handle never dialled
+
+	var connector driver.Connector = dsnConnector{dsn: dsn, drv: drv}
+	if dc, ok := drv.(driver.DriverContext); ok {
+		if connector, err = dc.OpenConnector(dsn); err != nil {
+			return nil, err
+		}
+	}
+	return sql.OpenDB(&countingConnector{Connector: connector, opened: opened}), nil
+}