No, there cannot be another triangle with the same side lengths but different angles.
This is because if all the sides of the triangle are 1 unit, then it is an equilateral triangle because all its sides are of equal length. A property of an equilateral triangle is that each angle equals 60°, because 180 ÷ 3 = 60, so it would be impossible to have an equilateral triangle all with side lengths of 1 unit that didn't have 3 angles of 60°.
I hope this helps!