Polyploidy is common in plants than in animals because in animals sex determination mechanism involves number and type sex chromosomes. Polyploidy will interfere with this mechanism and hence it is seen rarely in animals. I believe its probably because an animal progeny with multiple sex chromosomes won't be able to reproduce more of its kind unless it can also reproduce asexually (which is seen in plants as mentioned below).
Plants, on the other hand do not have any such sex determination (based on number or type of chromosomes) and majority of them can also reproduce vegetatively. This vegetative propagation enables multiplication of polyploid plants. Sexual reproduction will result in loss of desired polyploidy as per the principle of segregation. However, if a polyploid plant can produce a viable progeny sexually, it might have a different ploidy than its polyploid parent which won't happen if the parent reproduces asexually. Hence, polyploid plants can be classified as sterile and fertile polyploids (the reference will give a detailed account on this mechanism).
Refer the section of Polyploidy (page 115, Chapter 6) of this book (Principles of Genetics, Snustad and Simons, 6th edition)
When it comes to gene dosage, animals do show gene compensation of X-linked genes by various methods. Example- formation of Barr Body in human females (Note that this has nothing to do with polyploidy, but happens since females have one X chromosome more than males). Whereas gene dosage in polyploid palnts has been exploited for improved agricultural yields as they have shown better productivity than their diploid couterparts.