Yes, the terms Mycoplasma and PPLO refer to the same kind of organism. Pleuropneumonia-like organism, or PPLO for short, was the first name used to describe them in the 1930s (1). Mycoplasma is the genus of these types of organisms. This term was officially adopted in 1958 (2). Both names refer to bacteria that lack cell walls.
The name pleuropneumonia-like organism comes from the first one discovered. Its most recent name is Mycoplasma mycoides subsp. mycoides. This microbe causes a disease called contagious bovine pleuropneumonia (3). As more organisms of its kind were discovered, scientists agreed to refer to them as PPLOs. It is worth noting that not all PPLOs cause pleuropneumonia, though several species are pathogenic.
The word mycoplasma was originally used in 1889 by A.B. Frank. It described the “fungus-infected protoplasm” of a plant cell (2). In 1929, Julien Novak used the term to better classify the organism that caused bovine pleuropneumonia. He felt that
The name mycoplasma seems suited . . . to its unusual protoplasmic nature. (2)
Since Mycoplasma lack rigid cell walls, they can contort into different shapes. Other bacteria have a fixed shape. This flexible nature is what Novak was referring to. Eventually, Mycoplasma became the accepted name to refer to all PPLOs.