Basically, I want to have my bot on different pages using iframes and Bot Framework Web Chat.
Depending on what page the user opens the bot, a different dialog is called
Take a look at the backchannel. In your scenario you can pass a parameter from the Javascript via the back channel to set some value in UserData. Now your parameter will exist in state (in this case the IDataBag UserData) when your user first hits the bot.
Here is a C# sample of a 2-way backchannel
another C# sample using a 1 way backchannel