Two different problems get mixed up. One is keeping the American number your bank already has. The other is getting data, or a local voice line, once you are in Ecuador.
Wi-Fi calling
Apple’s guide describes Wi-Fi Calling as voice calls over a Wi-Fi network when the cellular signal is poor. On an iPhone it is turned on under Settings, Cellular, then Wi-Fi Calling. The carrier has to offer it. Apple says it is not available with every carrier or in every country, and that you should ask the carrier. When it is on, “Wi-Fi” shows after the carrier name, and calls use the Wi-Fi network. If the Wi-Fi drops, the phone can hand the call to the cellular network when that network is available.
A local SIM
A SIM bought in Ecuador is a local number and a local data plan. It does not keep your US number unless the US line is still in the phone, on a second SIM or over Wi-Fi calling. Ask the shop what identification it wants. This page does not invent a registration rule.
An eSIM for data
Airalo, the tracked partner already used on this site, sells a data eSIM. It is a way to be online. It is not a US voice number, and it does not replace Wi-Fi calling. Install it while you still have another connection. The plan size and the price are on Airalo’s page.
Banks and the IRS text codes to the number they have on file. If that number stops receiving texts, the banking chapter gets harder. Mail is the other half, in the mailbox chapter.