I just realized today that actually 2D beings with a full digestive tract are possible. The creature will not fall apart because of its disconnectedness.
Yes, you heard me right. Here's how this is possible: just because two 2D objects are topologically disconnected, doesn't necessarily mean they can be physically separated. For example, two concentric circles are topologically separate, but the inner circle can never escape the outer circle. Now, connect the inner circle to the outer circle with a narrow joint, then cut the outer circle on either side of the joint. Now you have a circular arc with two small gaps leading into the space between the outer circle and the inner circle. Assuming the circles are rigid enough, the inner circle will never escape the outer circle, so the space between them serves as a food pipe with an entrance and an exit. Behold! A 2D being with a complete digestive tract!
And yes, the 2D being is topologically two, but it's physically one since the two pieces are inseparable. Its two pieces can communicate with each other via signalling chemicals sent into the digestive tract where they can diffuse to the other piece. Of course, a pure circular layout introduces nitpicky problems such as the inner piece rotating relative to the outer piece, but that's not a problem if you pinch the digestive tract in the middle so that instead of a circular curve, you have a wavy-shape that keeps the inner piece always in the same relative orientation to the outer piece.
Furthermore, there's no reason to stop with just two pieces. Once can easily imagine a network of interlocking pieces, that communicate with each other by sending/receiving signalling chemicals into/from the gaps in between. In fact, this kind of interlocking structure is probably required to have an organism at all, since in order to have cells with a porous membrane for communication, the membrane would have to be a similar kind of interlocking surface, otherwise it would just fall apart.
The nice thing about this kind of interlocking is that the shapes of the various pieces can be adjusted to control the maximum width of the gaps between them, thus allowing only particles of limited size to enter the channel. You can even have a shape-recognition mechanism: imagine a wall with a circular gap in the middle with a rotating disk trapped inside. A certain section of the disk is cut out with a jagged outline. Only when something of the right shape comes along can it mate with the cut disk to form a full disk, which can then rotate to the other side and then detach, releasing the shape to the other side of the wall. If the wrong shape comes along, it won't fit and the thing won't rotate, so the wrong shapes are blocked from crossing the barrier.
This kind of mechanism can be used to form semi-permeable membranes, which are important to build a functional organism.