This Reindeer Hide Rug Care says
If looked after properly a reindeer hide rug ought to last for many years. At first the hide will definitely moult by shedding a little of its fur and this should be regarded as being normal. All you will need to do is carry it outdoors and shake the hide every few weeks. After 2 or 3 weeks the hairs on the hide should settle down somewhat and one will need to shake this outside the house more infrequently.
But it goes on to say
For general maintenance you may gently vacuum the hide but should use a brush accessory to delicately massage the hide rather than a smaller suction attachment as this could result in too much hair loss.
It is in all probability just as good to take the rug or carpet outside and shake it in the conventional way. It ought to be noted as with almost all fur pile goods you will certainly experience some hair loss from these hides.
So perhaps you've been hoovering the rug just like any other carpet, but it requires more delicate treatment.
One option is to bin it, and get another but follow the care notes better.