January in Novgorod (capital of the Kievan Rus) has average high/mean/lows of -5.9/-8.9/-11.6 celsius while Ulanbataar (current capital of Mongolia) has January averages of -16/-21/-26 celsius. Mongols were able to conquer 'Russia' in winter because the winters there were like Mongol springs (March temp ranges from -1 to -14).

The Mongols were simply used to colder. Mongolia is directly south of Siberia and cold winds make for some hellish winters despite their being on the same latitude as central Europe and the US. This was their reason for success and why they preferred campaigning in winter, while it was too cold for most armies, to the Mongols it was practically spring time.

To combat the cold they wore multiple layers and/or sheepskin/fur and protected exposed skin by rubbing animal fat on them for insulation.

The horses didn't need anything. They were capable of surviving Mongol winters in the wild and always lived outdoors, letting them look for grazing and fend for themselves in the cold.