i really liked Moldova in 2022! https://www.youtube.com/watch?v=bGNT5Uh-WKw
Deep longing is close, but it's more nuanced, and it doesn't have to be longing for someone, it can be someplace or activity, it also involves a kind of active recollection where you try to relive stuff in your memory, and there's an element of hope that you will experience whatever it is you miss again.
It's not a perfect translation, as you can "miss a call", "miss a good opportunity", etc and it doesn't mean that you are feeling saudade of it. But this case would be almost ok, just like we don't have a single word, for example, for the verb "to handle", which is a word that has almost the same idea in several contexts but we need a different word for each one ("lidar", "manipular", "mexer", "operar", "cuidar", "tratar", ...)
However, "to miss" is a verb and would be a translation for "sentir saudade". There's still no precise word for the noun "saudade", the feeling itself. Perhaps the closest one is "nostalgia", but it's still not exactly the same as nostalgia has this sense of something that happened in a distant past, and it is more related to the past itself than a person, a thing, etc. Even in Portuguese we have the word "nostalgia", but no one says "estou sentido nostalgia de você", we say "estou sentindo saudade de você". And we can feel saudade of something that happened yesterday, but not nostalgia.
Every real world usage of the term I can think of has it in the verbal form: "sentir saudade", "estar com saudade", "dar saudade", "a saudade é grande". Even in the sentence "Saudade" the verb is implied.
In all those cases it can be accurately translated as "to miss".
The same frequency as any other noun of feeling, like happiness, sadness, hate, etc. Saying "to miss" may imply that one is feeling that emotion but doesn't have a name to it, like having the emotion but not having the word that describes it.
Even though there are some ways to describe that idea it needs an extra effort, so it's not well defined as in Portuguese. Imagine not having a word for "responsibility", "power", "partnership" or other abstract concept: you can define them through subordinate clauses but it will not define that concepts (that's why have words).
And just for fun: if Inside Out was made in Brazil I can see clearly that Saudade would be a character (and a major one, the saudade of home is what leads to the climax), but it can't exist because it wouldn't have a word for that. In the second movie we have Nostalgia, but it's a minor character which wouldn't do what happened in the first movie (thus, we can see the difference)
Brazilian here. Not only a person, it can be a place, a thing, a scent, a moment and so on. It's kind of a nostalgia, but it doesn't need to be exactly related to a distant time. For example, if you are traveling you can feel saudade of your home the moment in the first day. You may also feel saudade when you listen to a song that you used to listen in a good moment of your life.
Nowadays everyone knows that it's a word that only exists in Portuguese (and in Galician, which is really close to Portuguese), but it was really shocking for me and other people that almost no other language has a similar word. Perhaps it is one of the most used words for describing a feeling here, it's like not having a word for "love", "anger" or "envy".
Other unique thing in Portuguese is that we have two verbs for the verb "to be": "ser" and "estar". "Ser" is when something always "is" and "estar" is for something that "is" at this moment. For example: if you translate "she is beautiful" as "ser" you expect that she is beautiful everyday, in all situations, while if you translate as "estar" is because she is looking good at this moment (perhaps she is wearing a nice dress, or recently has her hair cut, etc). Only having this in Portuguese was also shocking to me, as that distinction is natural for us.
Brazilians (my wife included) describe saudade as a sort of deep longing to be back with a loved one in person. Not sure if it's used in all Portuguese-speaking places or just Brazil (or where the term originated for that matter). The translation honestly seems straightforward enough but often when you ask someone to translate, they get that "on the tip of the tongue" look on their face and say English doesn't really have the right words for it.