That's a "wobble" board, introduced to UK audiences in the sixties asnd seventies (along with the Stylophone and didgeridoo) by Australian artist/entertainer/singer/songwriter Rolf Harris. "Tie Me Kangaroo Down, Sport" was one of numerous songs from his pen. He's still around, appeared at the Sidmouth Folk Festival some years back, and was more recently honoured? by being permitted to paint a portrait of HM the Queen.
I don't think linoleum would work - at least, not the stuff we used to get over here - too stiff and "dead". The sheet material used by Rolf Harris is known as hardboard, reconstituted wood fibre, smooth one side and textured the other. I think wobble-boards were even sold in music shops for a while.