Both other answers provided are quite good, but I'd like to address what I believe is the root of your misunderstanding:
A mortgage is just a loan, that has collateral attached to it. In the case of a mortgage on a house, that house is collateral, meaning the bank has some rights to that property if you fail to meet your mortgage payments. Many people tie their understanding of the mortgage, to the house sale in particular. In fact, you should consider it as two separate transactions: (1) You take out a loan from the bank, equal to the value of the mortgage; then (2) You pay the amount of the loan to the house seller [the bank will do that transfer to the seller's bank directly, because they do not want the risk of giving you so much money in cash].
Because a mortgage has collateral, it has lower interest rates than other types of credit - because it is less risk to the bank. If you have a mortgage on the home you live in, the bank feels you are less likely to just walk away from your obligations, because (1) you would be losing the value of the house; and (2) you are personally invested in living there. Because of #2, a mortgage on the house you live in, will be lower risk to the bank than the mortgage on a rental property (as pointed out by @NathanL).
So forget for a moment the second house you want to buy. If you want the bank to loan you $400k [80% of the value of your house], you could 'remortgage' your current home. The bank will regain the collateral of your house, meaning you are a low risk for them, and they will give you money at an interest rate generally similar to if you were just buying it new.
Now to your specific question: Will the bank decrease my interest rate, if I own another valuable property?
The answer is yes, if you give the bank collateral of that valuable property. It is the collateral they care about, not just the fact that you own it. It is true that having wealth will generally make you lower risk to the bank, but really what they want is the direct rights to something more valuable than your loan, should you default.
Will that lower interest rate be close to 0%?
No, because the bank still needs to make money. They just don't need to worry as much about you running away from your obligations, so they won't charge you as much of a 'risk premium'.