Popup calendar instead of date text field Adobe Sign forms

The pop-up date calendar is a relatively new addition. It's only available in Acrobat or Reader DC.

What application, including exact version, are your users using to open the file?

Acro Forms and Sign form are 2 different beasts. Sign forms have no support for any kind of JS, which is what is needed to support things like a calendar pop-up in Acrobat.

In Sign either signers enter a date into a text field, though the text field can be set to validate only for date formats.

In acrobat name the field like so

MyDate_es_:signer1:isdate(format=dd/mm/yyyy)

Sign will turn that into a text field called Mydate and assign it to th first participant and with a date validation in the dd/mm/yyyy format) 

Sign also has field to populate the current date in which case name your Acro field

Today_es_:signer1:date

This is an auto date which will be formatted per Sign account setting.