is it possible to import a shapefiles from manifold to Qgis with the same projection?
Shapefiles don't have projection metadata, they were designed only for use with longitude/latitude data - though many systems recognize an auxiliary .prj with WKT. Manifold forces you to export the .prj manually - but still you must make sure your drawing does not have a local scale or offset values, since WKT can't represent that.
Try it with Mapinfo format which is a vastly better format anyway. It's known as MIF, with .mif/.mid extension for the two files.
(Manifold does not yet support GeoPackage which I would otherwise recommend).
Better options also exist to transfer data via a shared database.
If you still have problems with MIF I'm happy to help find a way that works, there's lots of other options.