That's not necessarily true. My player has 3 citizenships but not eligible to play for one of those countries under FIFA rules.
My experience is with an under 18 player (minor). It's a formal application to FIFA that can only be prepared by a country's federation (e.g. FFA, FA). Solicitors do the application and they are only going to do it if the player has been on their radar and the youth nat team coach wants to select them for an international qualifying tournament or youth nat team camp. You have to show proof of nationality as well as living, studying and being connected to the community in that country for at least 5 years. Once they receive the application FIFA then contacts the other countries' federations and they need to respond that the player never represented their country, etc. FIFA reviews and then approves/rejects the application. The process took several months for my player. Not sure how it works for older (over 18) players (e.g. Sofia Huerta, Sydney LeRoux).