Vb6 "Tag" property equivalent in ASP.Net?

No, there's no direct equivalent, but if you're using v3.5 of the Framework, you can add this functionality quite easily using an extension method. For example:

Imports System.Runtime.CompilerServices

Public Module Extensions
  <Extension()> _
  Public Sub SetTag(ByVal ctl As Control, ByVal tagValue As String)
    If SessionTagDictionary.ContainsKey(TagName(ctl)) Then
        SessionTagDictionary(TagName(ctl)) = tagValue
    Else
        SessionTagDictionary.Add(TagName(ctl), tagValue)
    End If
  End Sub

  <Extension()> _
  Public Function GetTag(ByVal ctl As Control) As String
    If SessionTagDictionary.ContainsKey(TagName(ctl)) Then
        Return SessionTagDictionary(TagName(ctl))
    Else
        Return String.Empty
    End If
  End Function

  Private Function TagName(ByVal ctl As Control) As String
    Return ctl.Page.ClientID & "." & ctl.ClientID
  End Function

  Private Function SessionTagDictionary() As Dictionary(Of String, String)
    If HttpContext.Current.Session("TagDictionary") Is Nothing Then
        SessionTagDictionary = New Dictionary(Of String, String)
        HttpContext.Current.Session("TagDictionary") = SessionTagDictionary
    Else
        SessionTagDictionary = DirectCast(HttpContext.Current.Session("TagDictionary"), _ 
          Dictionary(Of String, String))
    End If
  End Function
End Module

Then, in your ASP.NET pages, first bring your extensions into scope, e.g:

Imports WebApplication1.Extensions

...and then use it your controls as desired:

TextBox1.SetTag("Test")

Label1.Text = TextBox1.GetTag

LATER EDIT: and if you really, really don't want to store your tags in the Session object, it's possible to stuff them into your Viewstate instead. This will of course mean that your tags will be exposed in the page markup sent to the user (albeit in obfuscated form), and, unfortunately, that some reflection-fu is required, since the ViewState property of a Page is marked as 'protected' for some reason.

So, this code should pretty much be considered for entertainment purposes only, unless you actually like to raise eyebrows during code reviews:

<Extension()> _
  Public Sub SetTag(ByVal ctl As Control, ByVal tagValue As String)
    ViewState.Add(ctl.ID & "_Tag", tagValue)
  End Sub

<Extension()> _
  Public Function GetTag(ByVal ctl As Control) As String
    Return ViewState(ctl.ID & "_Tag")
  End Function

Private Function ViewState() As Web.UI.StateBag
    Return HttpContext.Current.Handler.GetType.InvokeMember("ViewState", _
                Reflection.BindingFlags.GetProperty + _
                Reflection.BindingFlags.Instance + _
                Reflection.BindingFlags.NonPublic, _
                Nothing, HttpContext.Current.CurrentHandler, Nothing)
End Function

FINAL EDIT (I promise...). And here's a way to get rid of the reflection: first, create a new class to expose the ViewState property with a usable protection level, then change your Code-Behind (.aspx.vb) classes to inherit that instead of Web.UI.Page, e.g.:

Public Class PageEx
  Inherits System.Web.UI.Page

  Friend ReadOnly Property ViewStateEx() As Web.UI.StateBag
    Get
        Return MyBase.ViewState
    End Get
  End Property
End Class

Now, in your extensions module, you can access this newly defined property as:

Private Function ViewState() As Web.UI.StateBag
  Return DirectCast(HttpContext.Current.Handler, PageEx).ViewStateEx
End Function

Still a bit of a hack, but much more acceptable than using reflection...

You can also use the composite pattern instead of using inheritence:

public class TaggedControl<TControl, TTag> : Control 
 where TControl : Control, new()
 { public TaggedControl() {this.Control= new TControl();}

   public TControl Control {get; private set;}
   public TTag     Tag     {get; set;}     

   protected override void CreateChildControls(){Controls.Add(Control);}
 }

 var textBox = new TaggedControl<TextBox, string>();
 textBox.Tag = "Test";
 label.Text  = textBox.Tag;