Animals are not recognized as getting osteoporosis. This may be because nobody is looking. Animals are not routinely scanned for bone density the way older people are.
Horses are recognized as getting osteopenia, but not osteoporosis. Some veterinarians have claimed to have found enough loss of bone mass in the coffin bone of old horses' feet to merit the designation of osteoporosis.
Generalized osteoporosis appears to be rare in horses, but localized osteoporosis appears more common, especially in limbs that are not used.
Dogs (especially) and cats get arthritis, but they do not get osteoporosis. You might see some websites claiming dogs get osteoporosis, but this is not widely accepted among veterinarians. Again, it may be that pets are not examined as closely as humans, especially for outwardly invisible diseases like osteoporosis.
Scientists have studied bone physiology in bears because they spent large periods of time immobilized in hibernation. Immobilization is a contributing factor in the formation of osteoporosis, which is why people in wheelchairs often get the disease. Bears produce more parathyroid hormone during hibernation than during normal time. This hormone stimulates the growth of bone, and clues from bears may contribute to therapies for humans.