Exactly right. ; Basically, P&S cameras have a crop factor too - it's actually quite a bit more, usually on the order of about 6x. ; That '336mm equivalent' lens you had before was probably labeled something like 5mm to 60mm on the front of the lens. ; The equivalence was reached by taking the maximum 60mm end of the lens' reach, and multiplying by the nearly 6x crop factor of the tiny sensor.
Similarly, with any APS-C DSLR, SLT, or mirrorless camera, you're going to have a crop factor involved. ; All interchangeable lenses are labeled in actual focal length, which would be accurate when mounted on a full frame sensor. ; But when mounted on an APS-C sensor, you multiply by the 1.5x crop factor (Sony, Nikon, Pentax, Samsung...Canon is actually 1.6x on their slightly smaller APS-C sensors). ; So yes, that 55-210mm lens is giving you an equivalent reach of 82.5mm to 357mm in 35mm terms.
If you really want to stretch out the reach, you can do what I do and add a 1.7x teleextender to the end, and increase your reach all the way to 535mm equivalent - great for birding!