Formation of anatomical structures results from a combination of two kinds of process:

One is cell differentiation; and the other is geometric positioning of differentiated cells, so that the skin cells are on the outside, the muscle and skeletal cells on the inside, having the right shapes and connections to each other, and all 200-plus differentiated cell types located, shaped and interconnected in the patterns we teach and learn in anatomy courses.

How do all these different kinds of cells get put in the right places? They do it by a combination of two very different kinds of process: Let me call one of these Process A and call the other Process B, although we don't really have good names for either of the two.

By "Process A", I mean signaling to undifferentiated cells, according to their relative locations, which cell type they should differentiate into, for example stimulating cells located here, to become muscle cells and stimulate cells located over there to become bone cells, etc. etc. for all parts of the body. Most people think this is the essence of development, all there is to it.

But development also includes what I will call "Process B", which is for cell differentiation to cause cells to move to certain places; in other words for differentiation to control geometry.

This includes selective adhesion by membrane-attached proteins, including the different cadherins; it may include positive and negative chemotaxis. And perhaps galvanotaxis, which is alignment or guidance by electric field, to which tissue culture cells have been proven to respond. Proteins called "ephrins" have been discovered to induce breakage of cell-cell adhesions, thereby controlling gradients of eye brain connections and artery-vein separation. Many not-yet-discovered cell guidance mechanisms are ripe for you to find and study, which a good reason to focus on them.

Sponge and Hydra development consists of at least 90% Process B. (maybe 99%)
The locations where cells differentiate have little or nothing to do with their eventual location. Sponge cells constantly rearrange so much that their location today has little relation to where they were yesterday, or the day before (with the interesting exception of the earliest stages of egg development, which is quite orderly, not random.) Notice that Wikipedia says they "use reproductive cells" to produce buds. They are guessing, and are wrong. I think this mistake results from trying to explain this example of "Process B" development as if it occurred by "Process A".

Researchers are now trying to regenerate human arms and legs using what they call "stem cells", in the sense of tissue culture cells that have been prevented from differentiating. Do they assume that salamander legs regenerate by a spatially patterned differentiation of stem cells; or that embryonic human legs develop from masses of stem cells. Regeneration is by cell rearrangement, an example of Process B. Why do the expect to cause human legs to regenerate by a type A process? Because they haven't thought it through, and never bothered to learn enough about embryos. (Also there at least a billion dollars being bet on organ replacement by stem cells.)

Hydra budding is a re-arrangement phenomenon, and does not depend on growth. An analogy would be if someone took a wool sweater, unraveled a part of one sleeve, and then re-knitted that part in the shape of a branching tube. The tentacles, which are narrow tubes of the same differentiated cell types as the trunk, constantly add cells that had been parts of the trunk.

Imagine sleeves of a wool sweater constantly unraveling themselves and re-knitting much narrower tubes. (Or if one end of a vein split into a dozen capillaries)

Regeneration (in particular, regeneration of salamander legs) is mostly by means of process B.

Although muscles and cartilage cells appear to dedifferentiate, nuclear labeling studies consistently agree that when they re-differentiate, each cell changes back into the same cell type that they were before the amputation. For example, amputation through the elbow will result in shoulder chondrocytes rearranging to form a new radius, a new ulna, new carpals, and new phalanges.

(As many as half the chondrocytes in the regenerated radius, ulna etc. are derived from fibroblasts, but all the muscles of the new parts of the legs are made of cells that had already been muscle cells before the amputation. )

Blood vessel formation is (medically very important) example of cells rearranging themselves into hollow cylinders The Wikipedia article about "Angiogenesis" and "Vascularogenesis" are worth reading. The following is a quote from Wikipedia: "if a monolayer of endothelial cells begins sprouting to form capillaries, angiogenesis is occurring. Vasculogenesis, in contrast, is when endothelial precursor cells (angioblasts) migrate and differentiate in response to local cues (such as growth factors and extracellular matrices) to form new blood vessels. These vascular trees are then pruned and extended through angiogenesis."

The article defines these words as experts on these subjects currently do, dodging key issues. However, nobody even has a good hypothesis about the physics of biological tube formation.

I think they regard both vasculogenesis (vascularization) and angiogenesis as process B phenomena.

If you inject collagen into embryos, it will get wrapped tightly around arteries (Stopak & Wessells).