That's not what private property means here. It's private property from a business ownership sense yes but the store is in public for the purposes of personal privacy. The whole "reasonable expectation of privacy" thing is what governs your basic (keyword) right to photograph or record things and means more like your own house or a bathroom here, you have no reasonable expectation of privacy in a public store.

IKEA has no recourse to stop someone from filming. What they can do, however, is ask people doing this to either stop and continue shopping or to leave the premises. If they don't stop then IKEA can call the cops and have the recorder removed for trespassing. It seems like semantics but the reason you'd be in trouble with the police would be wildly different.