The last TeeNee trailer I had which still had a number on it had a decal on the frame on the right side. I can't say whether it was a serial number or just a model number, it was that worn. Some companies put the decals on the rear cross member as well. Most either don't have numbers, maybe were too old or the sticker is just long gone over all these years. Like mentioned above, your dealing with a trailer that's at least 35 or so years old. Maybe far more than that?
Just bought one with yellow paint underneath. Plate riveted on right side of crossmember with numbers. In 1955 Tee Nee Trailer Company of Youngstown, Ohio furnished around 90 percent of all trailer couplers used by the various trailer manufacturers in the country. This indicates that they dominated the market at the time. Tee Nee painted their trailers a yellow colour until 1962 when they changed to a "new metallic charcoal gray finish." Tee Nee's 1962 catalog showed eleven models ranging in capacity from 400 up to 3,400-pound boats.
I had a mid 70's era Tee Nee trailer here, about a 19' or so single axle galvanized model, the manufacturers tag and number were on the left side of the tongue, about 18" back from the coupler. It was barely readable from having the safety chains draped over it for years when parked, and from the tongue jack being bolted against it.