There have been conflicting reports on the electronic properties of twin domain boundaries (DBs) in MoSe2 monolayer, including the quantum well states, charge density wave, and Tomonaga-Luttinger liquid (TLL). Here we employ low-temperature scanning tunneling spectroscopy to reveal both the quantum confinement effect and signatures of TLL in the one-dimensional DBs. The data do not support the CDW at temperatures down to ~5 K.