fix(input-date-picker): hard to reproduce numbering-system caching issue (#8518)

**Related Issue:** #7958

## Summary

Resolves an extremely hard to reproduce caching issue that occurs due to
Alert's timeouts. The fix is to cache Alert's number formatter per
component so it doesn't impact other components.

Author: benelan

packages/calcite-components/src/components/alert/alert.tsx

@@ -31,7 +31,7 @@ import {
   connectLocalized,
   disconnectLocalized,
   NumberingSystem,
-  numberStringFormatter,
+  NumberStringFormat,
 } from "../../utils/locale";
 import { onToggleOpenCloseComponent, OpenCloseComponent } from "../../utils/openCloseComponent";
 import {
@@ -169,10 +169,17 @@ export class Alert implements OpenCloseComponent, LoadableComponent, T9nComponen
   connectedCallback(): void {
     connectLocalized(this);
     connectMessages(this);
+
     const open = this.open;
     if (open && !this.queued) {
       this.calciteInternalAlertRegister.emit();
     }
+
+    this.numberStringFormatter.numberFormatOptions = {
+      locale: this.effectiveLocale,
+      numberingSystem: this.numberingSystem,
+      signDisplay: "always",
+    };
   }
 
   async componentWillLoad(): Promise<void> {
@@ -201,12 +208,6 @@ export class Alert implements OpenCloseComponent, LoadableComponent, T9nComponen
   }
 
   render(): VNode {
-    numberStringFormatter.numberFormatOptions = {
-      locale: this.effectiveLocale,
-      numberingSystem: this.numberingSystem,
-      signDisplay: "always",
-    };
-
     const { open, autoClose, label, placement, queued } = this;
     const role = autoClose ? "alert" : "alertdialog";
     const hidden = !open;
@@ -265,7 +266,7 @@ export class Alert implements OpenCloseComponent, LoadableComponent, T9nComponen
 
   private renderQueueCount(): VNode {
     const queueNumber = this.queueLength > 2 ? this.queueLength - 1 : 1;
-    const queueText = numberStringFormatter.numberFormatter.format(queueNumber);
+    const queueText = this.numberStringFormatter.numberFormatter.format(queueNumber);
 
     return (
       <div
@@ -402,6 +403,22 @@ export class Alert implements OpenCloseComponent, LoadableComponent, T9nComponen
   @Watch("effectiveLocale")
   effectiveLocaleChange(): void {
     updateMessages(this, this.effectiveLocale);
+    this.numberStringFormatter.numberFormatOptions = {
+      locale: this.effectiveLocale,
+      numberingSystem: this.numberingSystem,
+      signDisplay: "always",
+    };
+  }
+
+  @State() numberStringFormatter = new NumberStringFormat();
+
+  @Watch("numberingSystem")
+  numberingSystemChange(): void {
+    this.numberStringFormatter.numberFormatOptions = {
+      locale: this.effectiveLocale,
+      numberingSystem: this.numberingSystem,
+      signDisplay: "always",
+    };
   }
 
   @State() hasEndActions = false;