Floating strip Micromegas detectors with one-dimensional readout strips are high-rate capable particle detectors with excellent spatial and temporal resolution, allowing single particle tracking for particle fluxes up to 7\,MHz/cm$^2$. A floating strip Micromegas detector collects the amplified ionization charge on copper anode strips being on high voltage, so called floating strips. The charge signal is read out by readout strips, separated by a thin insulating layer from the anode strips. This scheme strongly suppresses the sensitivity of the detector to discharges between the micro-mesh and the floating anode strips, induced by strongly ionizing particles. A novel two-dimensional readout has been realized with two layers of readout strips, parallel and perpendicular to the floating strips. We present results from characterizing measurements using a 20\,MeV proton beam. Charge signals were recorded with APV25 front-end boards, allowing for single strip readout with pulse height and timing information. To investigate hardware multiplexing schemes, advantageous for large area detector systems, we have interconnected groups of readout strips to single electronics channels, studying the signal pulse height. Additionally inclined detectors allowed for testing \textmu TPC reconstruction in both readout layers. We also report on alternative designs of the two-dimensional floating strip anode, where readout strip widths and geometry of both readout layers within the PCB have been varied, and its tests with 5.9\,keV X-rays emitted by a $^{55}$Fe source as well as with 22\,MeV protons.