Correct Photoshop does not have native svg support it a pixel editor not a vector editor. You can Place in a .SVG files as a vector smart Object there will not be any vector or multiple layers for the svg object just a rendering of the svg image is available. However, when you scale a vector smart object vector graphics will be use in the scaling process by re-rendering the scaled image by process the objects svg file.
There is a third Party extension available that is called SVGlayers. That can Import and export .SVG files. All SVG features are not supported and it performances is not very fast for it implemented via Photoshop scripting.
Its not expensive and you will no get the AI layer names you basicly get shape layer withe sequebcial layer names.