A rhombus is the only shape besides a square that can have equal sides.
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A Square
No, both shapes have all four sides equal.
Some shapes having four equal sides include rhombus, square, diamond, parallelogram with four equal sides, and oblique-angled equilateral parallelogram.
An infinite number. A polygon with any number of sides can be equilateral (ie have sides of equal length).
There are many such shapes and there is no specific name.