Why don't the stars in a binary star system of equal masses always orbit their center of mass in a circular orbit? If I have a binary object system of equal masses, why don't they always have the same circular orbit around their center of mass, like in the photo?
There's no reason why 2 bodies of equal mass couldn't have elliptical orbits around each other.   There's an example of that here

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The simple way to think about this is, if two bodies of similar mass approach each other, one of two things can happen, they either have sufficient velocity to pass each other with some hyperbolic curving of both objects or there's enough gravitational attraction that they capture each other in orbit, but the shape or eccentricity of the orbit depends on the ratio between the tangential velocity and the orbital speed at the closest pass.   For a circular capture the ratio needs to be exactly 1, and that kind of exactness is rare, so we nearly always get an elliptical orbit with an orbital capture.   

Solar-system planet or moon formation (not moon capture) tends to be much more circular, because elliptical orbits of objects tends to cancel out as large bodies coalesce, but you're never going to get 2 similar mass objects orbiting each other in a condensing gas cloud.   Most of the mass inevitably collects in the gravitational center.  There can be, however, a tidal influence that circularizes orbits over time, never reaching a full circle, cause that's impossible, but slowly becoming more circular.