It would be more accurate to say that most mathematical CS fits inside of constructivist foundations. Of course it also fits inside of classical foundations. So someone with constructivist inclinations may be drawn to that field. But participation in that field doesn't make you a constructivist.

As for what exists means, here are the three basic philosophies of mathematics.

The oldest is Platonism. It is the belief that mathematics is real, and we are trying to discover the right way to do it. Ours is not to understand how it is to exist, it is to try to figure out what actually exists. Kurt Gödel is a good example of someone who argued for this. See https://journals.openedition.org/philosophiascientiae/661 for a more detailed exploration of his views, and how they changed over time. (His Platonism does seem to have softened over time.)

Historically this philosophy is rooted in Plato's theory of Forms. Where our real world reflects an ideal world created by a divine Demiurge. With the rise of Christianity, that divine being is obviously God. This fit well with the common idea during the Scientific Revolution that the study of science and mathematics was an exploration of the mind of God.

Formalism dates back to David Hilbert. In Hilbert's own description, it reduces mathematics to formal symbol manipulation according to formal rules. It's a game to figure out what the consequences are of the axioms that were chosen. As for existence, "If the arbitrarily posited axioms together with all their consequences do not contradict each other, then they are true and the things defined by these axioms exist. For me, this is the criterion of truth and existence." See page 39 of https://philsci-archive.pitt.edu/17600/1/bde.pdf for a reference.

In other words if we make up any set of axioms and they don't contradict each other, the things that those axioms define have mathematical existence. Whether or not we can individually describe those things, or learn about them.

Over on the constructivist side of the fence, there are a wide range of possible views. But they share the idea that mathematical things can only exist when there is a way to construct them. But that begs the question.

Finitism only accepts the existence of finite things. In an extreme form, even the set of natural numbers doesn't exist. Only individual natural numbers. Goodstein of the Goodstein sequence is a good example of a finitist.

Intuitionism has the view that mathematics only exists in the minds of men. Anything not accessible to the minds of men, doesn't exist. The best known adherent of this philosophy is Brouwer.

My sympathies generally lie with the Russian school, founded by Markov. (Yes, the Markov that Markov chains are named after.) It roots mathematics in computability.

Erret Bishop is an example of a more pragmatic version of constructivism. Rather than focus on the philosophical claims, he pragmatically focuses on what can be demonstrated constructively. https://www.amazon.com/Foundations-Constructive-Analysis-Err... is his best known work.