What does it mean to be a constructivist/finitist?

Often I will hear constructivism and finitist phrased as a philosophical position rather than as a field of mathematics. I feel like phrases like "constructivists reject LEM" and "finitist reject the axiom of infinity" more often than "in constructive mathematics we don't use LEM" or "in finite mathematics we don't use the axiom of infinity" Similarly I never hear anyone say anything like "real analysts reject the fundamental theorem of algebra" Is there any reason for this? Is constructive mathematics / finitist mathematics not taken all that seriously?