Tbf, .223 and 5.56 are both .22 caliber rounds, as is .22lr, .22 short, .22 mag, 5.7x28, .22 nosler, .22-250, .224 valkyrie, .223 Wylde, etc.

Calibre (in this case .22) describes the bore diameter in inches (in this case .22 of an inch) or 5.6mm in metric.

Also 5.56 has higher pressures and a slightly different case (but it'll fit) than .223, they're not technically the same thing, though you can shoot .223 out of a 5.56 chamber. Not recommended to shoot 5.56 out of a .223, but tbh in most modern guns you'll prob be fine (can't sue me if your shit blows up.)