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Why do I need to provide a mailing address and phone number in order to add a credit card to my account?
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To verify its authenticity.
It is not necessary information to verify the authenticity of a credit card, and it exposes the user to greater risk. There is no explanation of how the information is to be used or protected or for how long it is going to be kept.
DigitalOcean is asking you for your billing address to verify that you own the credit card you provided to them. Ultimately, the risk of constant credit card fraud by scam artists outweighs the loss of privacy here.
It might be worth asking support how they store the info, and if they keep it, but there's nothing out of the usual going on here: most notable VPS providers and online stores do the same thing.