Pianos almost always have a serial number somewhere. They can be in unusual places and this page shows some of the places to find the serial number. Dates that are cast into the plate of a piano are dates when the company was founded and not dates for the particular piano. You are usually looking for a 5 or 6 digit number. Some piano have other numbers stamped on them that are used in the manufacturing process. For European pianos, Bill Kibby in England has better information than we do at Piano Gen. There is a fee for some of his services
The most common place on a grand piano is stamped on the plate in the V shaped area between the tenor and the bass sections of the plate or at the treble end.
Sometimes the number is stamped in the soundboard under the strings.
Kimball pianos sometimes have the serial number in a hard-to-find place stamped on the back edge of the keybed. This is looking from the tail end of the piano.
Uprights usually have the serial number stamped on the plate near the top of the piano or stamped in a window in the plate.