Reproductive budding and what you call 'cellular budding' are really highly related processes. Budding as a form of reproduction essentially partitions protein aggregates and damaged cellular components into the host or mother and builds fresh or 'young' cells on the opposite side of a partition. To begin understanding this look at Saccharomyces cerevisiae (budding yeast) which forms protein rings (from the septin proteins) at the membrane, around the bud neck which separates the mother and daughter cells Hartwell 1971. This ring acts a partition that in part, withholds protein aggregates and certain proteins from diffusing from the mother to the daughter. This protein ring is an example of how cells limit diffusion of proteins and cellular components to the daughter cell. Another good example that comes to mind is Linder 2007, though it is done in E Coli, not budding yeast, where mother cells maintain protein aggregates and age, while the daughter cells are given fresh components and are therefore more fresh and 'young'.
Now like you mention, imagine this process in a multicellular organism to be fundamentally the same. At some point the multicellular organism will start an outgrowth of cells, while restricting what materials are given to the daughter cells to maintain their youth. And eventually a new organism will have been created. Some of the details will be different, but the fundamental process is is quite similar. In that you start with an old cell that creates a new cell from scratch, but rather than splitting all cellular components equally between mother and daughter, the daughter cells is made in peak condition while the mother cell retains much of the cell 'junk' like protein aggregates.
Hopefully that starts to answer your question.