Occasionally, yes, coins were used as a source for silver: a rare example would be George Washington having a dozen small silver camp cups made from 16 silver dollars. But the word COIN stamped onto silver objects means that the silver was the same proportion as that used for coinage, or 900 parts per thousand as opposed to the higher 925 parts per thousand for the sterling standard. The remaining portion of the alloy was usually copper, to strengthen the otherwise too-soft pure silver. Sterling objects (like teapots or candlesticks) were sometimes melted down by silversmiths so they could make their customers new objects in the current, more fashionable style. These were marked as sterling, the 925 parts per thousand silver content.
When the American colonies belonged to England, silversmiths of course followed English laws in marking their silver. After independence, standards varied among the various states. In 1837, Congress passed a law that established 900/1000 as the official standard for coin silver.
See Elmer’s spoon up there? And see the word COIN stamped to the left of Elmer’s name? This let people know that the silver content in the spoon was the same as in the coinage, or 900 parts per thousand. Elmer’s spoon–and most other items similarly marked–were almost certainly not made from melted coins.