Do I need an unique key for each computer I use to access the same droplet?
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No, but it is good practice to keep your keys unique to each computer.
Say your laptop is stolen: just delete that key from your servers authorized_keys file and your done.. you still have access from your desktop, but the laptop can no longer access your servers.
Having said that: if you only have a laptop, and it crashes, but you salvage all of your files…you can just use the same key on your new computer without having to generate new keys and such. (this happened to me recently, all keys worked fine on my new desktop)