What type of stitch is a perfect fit for elastic materials (as close as possible to Nike's sport gear for running)?
Stitching a stretchy fabric requires a stretch stitch, to allow the seam to stretch with the fabric itself. Multiple types of stretch stitch are available to the home sewer, though some require specialized machines to achieve.

ZigZag Stitch

The back and forth motion of the needle in doing a zigzag stitch allows the seam to have some give and stretch with the fabric. The more back and forth, the more stretch, so shorter stitches will provide more flexibility. This can be achieved with any basic sewing machine, and is the most accessible.

Stretch Stitch

Some machines will have a "lightning bolt" shaped stretch stitch setting. This is similar to a zigzag, but with an extra step backwards to add even more give to the stitch. It will also look closer to a straight line than a zigzag. This is the next most accessible, as it's available on many, but not all, standard sewing machines.

Twin Needle Stitch

This stitch will require a special double needle to be added to your machine; you'll also thread it with both a spool and a bobbin of thread at the same time, separating the threads when you reach the needle. The upper layer will have a double row of stitching, while underneath the bobbin will have a zigzag format allowing the seam to stretch. This is likely the closest to what you're looking for, and can be accomplished with most standard sewing machines. (I cannot promise that the twin needles will be compatible with all machines.)

Overlock

Overlock stitching is what is often seen on the inside of commercially produced garments; it's a stitch with loops of thread wrapping around the raw edges of the fabric. It has a very large amount of stretch, but requires a specific machine (known as either an overlock machine or a serger), and is only useable on inner seams, as sergers also trim the edge of the fabric as they work. While the trimming can typically be deactivated, the edge wrap still leaves the machine only suitable for seams, not top-stitching. (As an anecdotal note, this is what I most often use for stretchy fabrics.)

Cover Stitch

This is likely what Nike and other companies are actually using. This requires a cover stitch machine or a convertible serger. The stitch produced is something of a cross between the twin needle and overlock stitches, and is suitable for hems and top stitching on stretchy materials.

You can see examples of these stitches and greater detail (as well as finding a link with details on twin needle stitching) here, which is also where I took the order of stitches and some technical details from.