Nope Saudade is just the feeling of nostalgia, for some reason someone convinced foreigners and some Brazilians that it’s special or somehow different from nostalgia. And it caught because mostly we don’t use nostalgia to refer to people, so in most people’s mind they’re somewhat different but it’s just because of common use. Any phrase that Portuguese uses saudades can be replaced with missing in english without losing any meaning, e.g. Tenho saudades do meu cachorro -> I miss my dog.