1. A monitoring device for monitoring one or more facility elements in a building, comprising: a laser range finder to measure a three-dimensional shape in a periphery of the monitoring device; a physical quantity sensor to measure a physical quantity of a facility element under monitoring from among the facility elements in the periphery of the monitoring device; a visible light camera to capture a visible light image of the facility element under monitoring in the periphery of the monitoring device; a memory storing instructions and storing a localization map including shapes and positions of structural objects inside the building other than shapes and positions of the facility elements, a facility element collation map including the shapes and positions of the facility elements inside the building, standard physical quantity data of the facility elements, schematic diagram data for the facility elements and connections thereof, and an installation error statistical data including a plurality of error distributions of the facility elements; a processor connected to the memory, the visible light camera, the physical quantity sensor, and the laser range finder; and a display unit connected to the processor, wherein the memory stores instructions that, when executed by the processor, cause the processor to: estimate a position of the monitoring device by collating the measured three-dimensional shape with the localization map including the shapes and positions of the structural objects inside the building other than the shapes and positions of the facility elements; extract the facility elements in the periphery of the estimated position from the facility element collation map including the shapes and positions of the facility elements inside the building; extract a shape and position of a facility element candidate from the measured three-dimensional shape; calculate a plurality of similarities between the shapes and positions of each of the facility elements in the periphery of the estimated position extracted from the facility element collation map and the shape and position of the facility element candidate extracted from the measured three-dimensional shape based on a plurality of error distributions of the shape and position of the facility elements in the periphery of the estimated position and an error distribution of the shape and position of the facility element candidate extracted from the measured three-dimensional shape; identify, as the facility element under monitoring, one of the facility elements in the periphery of the estimated position extracted from the facility element collation map that corresponds to the facility element candidate extracted from the measured three-dimensional shape based on the calculated similarities; calculate a three-dimensional physical quantity distribution based on the measured physical quantity and the measured three-dimensional shape; compare the measured physical quantity with the stored standard physical quantity data of the facility element under monitoring and determine whether the measured physical quantity of the facility element under monitoring is abnormal; calculate a three-dimensional visible light image based on the captured visible light image and the measured three-dimensional shape; create a superimposed image of the measured physical quantity of the facility element under monitoring and the three-dimensional visible light image, and convert the superimposed image into a two-dimensional image; and display the two-dimensional image and a schematic diagram including the facility element under monitoring and the connections thereof based on the schematic diagram data on a same screen of the display unit, wherein, when the measured physical quantity of the facility element under monitoring is abnormal, the facility element under monitoring is highlighted in the two-dimensional image and the schematic diagram.