Yes. By most definitions, Toki Pona is a language (and is real). It is a set of words and grammar rules that are used to communicate. Even though Toki Pona is designed to be small and limiting, you can talk about anything in Toki Pona with enough knowledge and perseverance. Toki Pona can be a written and signed language, and has its own full accompanying sign language, luka pona.
Constructed languages are a type of language, even though they are created artificially. All languages are "made up", just in different ways. In fact, many "standard" varieties of natural languages are constructed and artificial.
According to Noam Chomsky's theory of universal grammar, language exhibits recursion. Toki Pona arguably has forms of recursion, but uses anaphora instead of recursive relative clauses. The theory of universal grammar is notably controversial. Pirahã is claimed to be a natural language that lacks recursion.
Toki Pona may not be a living language as there are no known native speakers yet. In this respect, it is similar to languages that have died out, like Latin. There are well over a thousand nonnative speakers according to the Toki Pona census.