Only the home where you live most of the time is your primary residence. You might change your primary residence in mid-year (such as when the new house is finished) but you can only have one at a time.
You can deduct property tax on all real property you own in the US, up to the SALT cap. The property doesn't even have to be your main residence.
You can deduct mortgage interest on your main home and one second home. The home under construction can be your second home and you can deduct the interest on your construction loan, provided that you complete construction and move in within 2 years from starting, and that the construction loan is converted to a mortgage after construction is finished. (If it takes more than 2 years to build the home, or if you don't convert the construction loan to a mortgage, then the construction loan interest is not considered deductible mortgage interest.)
