Milk consumption and osteoporosis
In a large study in Sweden they observed an association between high milk consumption and increased incidence of osteoporosis, especially in women. The authors suggested that one possible explanation could be reverse causality: People who knew they were at increased risk of osteoporosis, for example, due to family history of osteoporosis, might drink more milk in an attempt to prevent osteoporosis, but they still developed it.
High milk intake was associated with higher mortality in one cohort of
women and in another cohort of men, and with higher fracture incidence
in women. Given the observational study designs with the inherent
possibility of residual confounding and reverse causation phenomena, a
cautious interpretation of the results is recommended.
Also, the evidence from several systematic reviews of studies in another answer shows that high milk/dairy intake is associated with a slightly decreased and not increased risk of osteoporosis and bone fractures.
Plant vs animal protein and osteoporosis
In addition to the review mentioned in another answer, in the following 2 study reviews, animal protein was not associated with increased rate of osteoporosis:
1) Animal versus plant protein and adult bone health: A systematic review and meta-analysis from the National Osteoporosis Foundation (PlosOne, 2018):
OBJECTIVE: The aim was to conduct a systematic review and
meta-analysis evaluating the effects of animal versus plant protein
intake on bone mineral density (BMD), bone mineral content (BMC) and
select bone biomarkers in healthy adults.
CONCLUSION: These results do
not support soy protein consumption as more advantageous than animal
protein, or vice versa.
2) Acid diet (high-meat protein) effects on calcium metabolism and bone health (USDA.gov, 2010):
On the basis of recent findings, consuming protein (including that
from meat) higher than current Recommended Dietary Allowance for
protein is beneficial to calcium utilization and bone health,
especially in the elderly.
In conclusion, there is no convincing evidence to say that milk or animal protein consumption is associated with an increased risk of osteoporosis.