Hi,
There is no such mechanism.
Now, for custom item fields with "values", like radios, dropdowns or checkboxes, It is possible to actually create a custom field of the table "product" where you can select the values to be used in each product and the custom item field will then use these values. To make that connection between both fields, the custom product field needs to have the same column name as the other with "_values" appended to it.
So if you wanted to allows the selection of a date among only a few dates for each product, it could be done like this, without using a date picker.
Now with a date picker, being able to override some of the settings of the field based on the product selected would require customizing the code of the plugins/hikashop/datepickerfield/ plugin code and creating custom product fields where to fill in these override values in each product. So it's not impossible, but it would require a developer to work on this for you.