Air carrier First Officer reported disagreement between the Captain and First Officer flight displays during descent in icing conditions resulted in altitude and airspeed deviations. Post flight inspection revealed layers of ice on probes.

Narrative

On initial climb conducting training in the CRJ 200. We initially climbed through 5000 ft. and returned to assigned altitude. Busy airspace in icing conditions conducting training contributed to altitude deviation. On descent airspeed exceeded 250 on descent. Used automation to reduce workload and slow aircraft to 250 knots. Aircraft was operating in areas of MOD icing; however no visual cues were seen on typical areas for ice accumulation. On approach airspeed on Pilot in Command's (PIC's) side was above bugged speed and First Officer (FO) was in the speed bug. Executed a missed approach/go-around. On go-around altitude was +/- 300 ft. and PIC assumed Pilot Flying (PF) responsibilities. PIC over-controlled aircraft assuming controls. AP & FD were giving incorrect guidance for modes that were being requested. PIC disconnect automation and flew aircraft raw-data back around for approach in IMC conditions. On approach altitude and airspeed were fluctuating -250 ft. & airspeed +/- 10 knots. Landed via ILS to visual after breakout. EFIS COMP MON HDG Flag was intermittent with associated caution throughout flight/missed approach. Taxied to gate without incident.PIC as Pilot Monitoring (PM) task saturation; instrument cross-check; and instruction given. FO as PF inadequate cross-check raw data with Flight Director guidance. PIC as PF over controlling aircraft; and unawareness of raw data disagreement between both PIC and FO displays. Weather consideration MOD icing reported in ZZZ below 6;000 ft. majority of flight conducted in icing conditions. On Post-Flight inspection First Officer noted probes had layers of ice adhered to them; and ice detector on First Officer side of aircraft had more ice accumulation.Better awareness of potential affects of icing on instrument readouts; cross side instrument cross-scan; consider deactivation of automation when information is confusing and does not agree; use the word 'UPSET' when aircraft is in known banking condition opposite the clearance instead of giving directions as PM towards the right course since this is the most concise way to achieve a shared mental model of what is occurring.