Cocktail shaker - silver plate but black on the inside
Yes, it's safe.

Silver tarnishes easily, particularly in contact with sulfur-containing foods, particularly egg. The tarnish is not hazardous, and can be cleaned off with silver polish or with DIY chemistry like in Tetsujin's answer. Black tarnish doesn't form readily on its own, so if you're not planning on using eggs in your cocktails then cleaning off the existing tarnish will likely last you a while (it'll be bright silver initially, but will quickly tarnish to a dull greyish silver). However, it can also be ignored (which was apparently the approach taken by the previous owner).