The best thing to do is take it to a dealer or collector for their opinion.
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Yes, Liberty Seated dollars were struck in 1865.
The first real person on a dollar coin was President Eisenhower in 1971. No silver dollar coins have portraits of real people.
If the coin is a real Morgan or Peace dollar. The value is just for the silver, about $20.00.
The most basic thing is the weight and diameter of the coin. if you have doubts about the authenticity of the coin, take it to a coin dealer for an assessment.
No U.S. silver dollars have an Indian Head design, so if you have one it's not real. A genuine silver dollar coin dated 1851 would be a Seated Liberty coin. I have seen many 'fantasy' silver rounds with the Indian Head design with different dates, but they all have the 1oz .999 fine silver markings somewhere on the coin.