Yes, this sounds natural, but we would not usually use the idea of 'close together' with feelings. A more standard way of saying this would be 'My feelings and those of my sister are so similar'.
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My feelings and those of my sister's are so close together.
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You could say:
- My feelings and those of my sister are very similar.
- My sister and I feel the same about ...
- My sister shares my feelings about...
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