A Pythagorean triangle is a right triangle with integer sides. A primitive Pythagorean triangle is one in which the sides have no factor in common. For example a triangle with sides (30, 40, 50) is a Pythagorean triangle but not a primitive Pythagorean triangle.
It is possible for two primitive Pythagorean triangles to have the same area. The smallest example is (20, 21, 29) and (12, 35, 37). Both have area 210.
It’s also possible for three primitive Pythagorean triangles to have the same area, but the smallest example is much larger. The triangles (4485, 5852, 7373), (1380, 19019, 19069), and (3059, 8580, 9109) all have area 13123110, discovered by C. L. Shedd in 1945.
Nobody has found an example of four primitive Pythagorean triangles having the same area. I don’t know whether it’s been settled whether such triangles exist. But it has been proven that if they exist, they have area greater than 9.3 × 1024. See OEIS A093536.
Incidentally, the triangle (20, 21, 29) came up in the post Do perimeter and area determine a triangle? from February of this year.