Muonium is an exotic atom consisting of an electron bound to an antimuon. Structurally, it is analogous to positronium; however, because of the muon's larger mass (210 times the electron's) muonium resembles hydrogen more closely than positronium. High mass = small wave packet in the subatomic world, so a muon is small compared to an electron. It is huge compared to a proton, of course, but still small enough so that it can be treated like a nucleus for purpose of studying atomic structure.
Muons, like electrons, are leptons. A muon can only decay into electrons and neutrinos. It is energetically impossible for a muon to transform into a proton. Muonium does not decay into any kind of nuclide, stable or radioactive. In that sense, muonium is very much not an element.
Its muon is the only unstable part of a muonium particle, so the half-life of muonium is the same as that of a muon itself. Muon mean life is (2.1969811±0.0000022)E-06 sec, which makes its half-life 1.52 microseconds.
It can participate in chemical reactions, forming compounds like muonium chloride.
Positronium is a member of a class of exotic atom known as an onium - a particle and its antiparticle bound together. Muonium, despite its name, is not an onium since electrons and (anti)muons are different kinds of particle. Muons should form an onium, which is referred to as "true muonium". This is a fine example of how terms sometimes just collide.