This claim is very false. Starbucks is not planning on phasing out dairy to meet climate goals, and Starbucks isn't banning milk anytime soon. In a Fortune article, Starbucks executives explained that their actual focus is on becoming more resource positive by storing more carbon than they emit, cutting down on waste, and providing more fresh water than they use/produce. Starbucks also talked about investing in renewable energy, which shows their goals are about sustainability overall, not removing dairy from the menu.
With so many milk alternatives available today, some people assume Starbucks will eventually drop dairy, but that isn’t what the company has said. A LiveKindly article pointed out that the World Wildlife Fund found a single dairy cow produces around 17 gallons of manure and urine every day, which releases greenhouse gases, pollutes water and air, and harms wildlife habitats. On top of that, it takes about 144 gallons of water to produce just one gallon of milk in the U.S., with more than 93% of that water going toward growing food for the cows. Each cow also eats close to 100 pounds of food daily, which adds to the environmental impact. Even with those numbers, Starbucks has not announced any plan to eliminate dairy.
Sources: https://www.livekindly.com/starbucks-phase-out-dairy-climate-change/
https://fortune.com/2020/01/21/starbucks-carbon-footprint-dairy/