Daniel Carl Muller was born in Germany in 1872 and arrived in the United States with his family in 1882. His father, John Heinrich Muller, was a carver and a personal friend of Gustav Dentzel; the elder Muller had worked alongside Charles Looff on the installation of the first carousel at Coney Island in 1876. When their father died, the teenage Daniel and his younger brother Alfred were taken into the Dentzel family and trained in the Dentzel workshop in Germantown, Pennsylvania, where Daniel emerged as the more inspired artist of the two. He pursued formal training at the Pennsylvania Academy of Fine Arts under the sculptor Charles Grafly, where he took classes intermittently for twelve years and won the Edmund Stewardson Prize for sculpture. He made frequent Sunday trips to the Philadelphia Zoo to sketch the exotic animals, and in 1898 designed a mermaid-themed fountain for PAFA that was cast in bronze in 1905 and installed in the academy's lobby, where it remains today. Despite his fine-art training, Muller ultimately chose carousel carving as his vocation.

In 1903, Muller and his brother opened D.C. Muller & Bro., advertising on their business card "Caroussels of high artistic merit." Through 1914 the firm produced approximately twelve to sixteen carousels, characterized by realism, dramatic anatomical proportions, and exceptionally refined military trappings and accoutrements — a sensibility shaped equally by Muller's PAFA training and by his disciplined zoological reference work. Unable to compete with the larger and more efficiently organized contemporary American manufacturers, the Muller shop closed in 1914; Daniel continued to carve for the Philadelphia Toboggan Company and other firms, and in 1918 rejoined the William Dentzel Company under Gustav's son. Only two D.C. Muller & Bro. carousels survive intact: the Forest Park Carousel in Queens, New York — a designated New York City landmark — and the carousel at Cedar Point in Sandusky, Ohio. The foundational carousel historian Frederick Fried regarded Muller as the greatest of all master carousel carvers, an assessment that has carried forward in the auction market, where Muller figures consistently command among the highest prices realized for American carousel work.