I'm seeing a lot of references to <PERSON> limits and the like in the answers here, but to me, that is implying a far slower process than what I would think would be described as a "collision". Just assuming a collision velocity no greater than that starting the two planets at rest and letting them fall together under their own mutual gravity, they're going to be going so quickly by the time they collide that there certainly won't be time for them to break apart due to being within the <PERSON> limit.
If we take this question as Earth colliding with Jupiter, then technically, theoretically, yes, Earth would begin to break apart, but we're talking about a minimum collision speed here of about 60 km/s (Jupiter's escape velocity). The entire collision at that speed, from first contact of atmospheres to centers-of-mass coinciding, is only about 20 minutes. So while technically one or both would begin to break apart, the differential acceleration between their nearest and furthest points will be on order of about 4 m/s², making the total breakup on order of about 4800 km, or around a third of the diameter of the Earth. Not insignificant, but fairly trivial in terms of the overall impact's effects. (Consider the difference between a cannonball splashing into a barrel of water vs. a dumbbell. 921
Yeah, technically they'll behave somewhat differently, but in terms of the level of catastrophe, it's pretty trivial.)
Far more important is to consider that at the magnitudes of forces and velocities involved in a collision like this, everything behaves as a fluid. So yes, the analogy of it being closer to imagining two spheres of water colliding than two solid objects is quite apt.
Interestingly, there's a theory that almost exactly this happened in Saturn's past: a roughly Earth-sized object may have collided with it in the distant past. This did not, of course, result in Saturn breaking up, but it may have resulted in 'splashing' its core into something quite a bit less orderly and more 'fuzzy' than suspected before. It may even still be 'sloshing' in a sense, creating some of the resonances responsible for the specific patterns of its rings!
I'm also seeing a lot of assumptions of velocities quite a bit lower than Jupiter's escape velocity, but... how would this happen? If we put ourselves in a Jupiter-centric frame of reference (but non-rotating), then the least-fast object that could conceivably collide with it would be one that is stationary at infinity, i.e. at its escape velocity. In order for a colliding object to be going slower than that, it would need to pop into existence at some finite distance from it, which can't really happen outside a simulation.
So, combine the minimum speed possible for this scenario (60+ km/s²), the short duration of the collision (1200 seconds or so), and the sheer forces and amount of heat involved, and what you'd have is more of a 'splash' of a smaller sort of egg-shaped ellipsoid of water hitting a much larger and more massive sphere of water.
For an intuitive sense of what it might look like, check out some simulations of the collision that is theorized to have created the Moon, such as: https://youtu.be/o2lRpiediP8?t=340 This would be quite a bit less dramatic, with 300:1 mass difference instead of 10:1, but still enough to hopefully give some idea. 921