That’s not at all how interest rates work on homes… They’re fixed and yes while fluctuations in interest rates can have an effect on the home value, that won’t change how much your mortgage payments are, and can only effect your property taxes by at most 2% per year

Rent doesn’t have to fluctuate with interest rates at all as it’s up to the home owner

Edit: unless you’re in a variable rate interest loan in which case, yeah your lender is screwing you