P.T. Barnum is a legitimate type face (sorry, but "font" is misused by the digital age folks), but it is not what is used on the D&S. For a reference, see the link. The lettering style the D&S uses is unique to this railroad as far as I know and evolved through railroad use rather than picking up a standard railroad lettering. It has the "old time" appeal, but is not really authentic in this usage. The very highly refined "railroad Romans," or even gothics, with appropriate shading, would be authentic. A review of old photos will show this, but the lettering on the D&S is of modern vintage, and had its origins in the 1950s on the Silverton equipment. The numerals used on the locomotives are well done and are authentic to D&RGW style, but tender and coach lettering are not authentic to anything prior to the 1950s.
P.T. Barnum is classified as a French Clarendon. It was originally cast as a metal printer's type by Marder, Luse & Company in the late 1800s, and this company was absorbed into the American Type Founders Company in 1892. It was released as a revival casting by ATF in 1933 where the name P.T. Barnum was first used for this design, and in 1949 ATF reissued the type again and carried it in inventory up to ATF's demise in 1993. It was commonly used from the 1870s onward, and was often used in railroad printing. I have at hand a ticket from the 1890s that has Silverton Railway Co. and Red Mountain & Ret. set in either 8 or 10 point French Clarendon (P.T. Barnum). But, as letting on the D&S equipment, this is the incorrect type face, or font if you're digitally inclined.
The casting matrices for several sizes of P.T. Barnum still exist and this type is still available as metal type.