What's an ideal wire? I'm not talking about an ideal wire in a circuit (a wire with infinite conductance).
I'm talking about an ideal wire in the case of the magnetic field of an infinite current carrying wire. What dimensions must an ideal wire have to better approximate an infinite current carrying wire in terms of its magnetic field?
For example, for a parallel plate capacitor, the field better approximates two infinite plates as $$A >> d$$ where $A$ is the area and $d$ is the length.
Or for a solenoid, the field better approximates an infinite solenoid as
$$L>> d$$ where $L$ is the length and $d$ is the diameter.
For a current carrying wire, what dimensions must it have to better approximate an infinite wire?
 A: Your question if I understood correctly is about the significance of the infinite length of a straight wire where the Biot Savart Law is concerned.
You will find that for a finite wire, the only area where the Law would hold is at L/2 along the horizontal length of the wire, i.e, the entire wire is horizontally symmetrical about that plane and thus the magnetic field on this plane is perfectly orthogonal to the wire and current flow. This symmetry is important as it allows the horizontal components from each current element in the wire on both sides of the center to cancel out (to produce an orthogonal magnetic field). What this means is that for anywhere in the plane where I have drawn the circle at L/2 (outside the circle also fine as long as in the same plane), the magnetic field will be predicted exactly by the Law. 

We can see why we prefer to talk about infinite wires where Biot Savart Law is concerned--so that any point (not just on one plane in the middle of the wire) we pick to measure magnetic field, it is pretty much always going to be L/2, with the wire horizontally extending equally far away from that point, and the Law will be useful.
