Why do people describe a swap as a series of forward-like cash flows?

A plain-vanilla swap has multiple settlement dates. On each date, the parties compare the fixed leg and floating leg and make a net payment. That is similar to having a sequence of forward commitments on future rates.

The notional amount is usually not exchanged in a standard interest rate swap. It is the reference amount used to calculate the cash flows. If the fixed payer owes 4.10% and receives floating of 4.60%, the fixed payer receives the net difference for that period, adjusted for the payment frequency and day-count convention.