The base cards have an old look to them, but I can't find a direct design from the past that they copied.
The set also has some SP variations and inserts that used the 1940 Play Ball design. I guess by including Play Ball, UD was indirectly ripping off Topps. The Play Ball set was put out by Gum, Inc. of Philadelphia. That is the same company that later put our Bowman cards and was bought out by Topps in 1955.
Vintage also includes some Old Judge cards, those none of the 1880s versions of Old Judge looked like this.
I don't know if the sales of Vintage couldn't complete with Heritage or if UD ran out of designs to steal, but the 2004 set was the final one.