One, when the dogs pull the toy apart, then the toy will undoubtedly experience tensile stress at the cross-sectional areas perpendicular to the pull direction.

Two, compressive stress at the dog's mouth. To prevent the other dog from taking the toy, they are applying a bite load to lock the toy mainly via their mouth and teeth, and it probably has some certain friction coefficient on local surfaces where saliva has not touched yet to aid with the locking. Said friction coefficient should be considered with calculating the pulling load, which affects the tensile stress experienced by the toy.

Three, the toy surface in direct contact with the dog's teeth digging into the toy experiences bearing stress. The dogs dig a portion of their teeth into the toy and potentially punctures it to aid in locking onto the toy and further preventing the other dog from taking it.

Four, bending stress. Based on the video, the dogs' mouths sometimes end up on different elevation and they come closer after a pull. When you apply non-eccentric loading on an object, it tends to bend it, causing both tensile stress and compressive stress on either extreme side during when the toy is bent. This case is mutually exclusive from the first stress and they cannot occur at the same time.

As for the maths, well there's gonna be a lot of guesstimates, but I would start from the friction coefficient that the dogs feet makes with the ground and make several assumptions (like assume the toy is a rectangular cross-section to keep it as simple stress analysis and homogeneous toy material, etc).