In most cases, a good secure muscle repair and adequate skin resection (plus the resultant internal scar layer that helps to reinforce the abdominal wall) at the time of tummy tuck will yield a good, long-lasting result. In some cases, intra-abdominal fat (mesentery and omentum) can contribute to trying to get the equivalent of a "ten-gallon head into an eight-gallon hat!" The abdominal contents are fitted into a tightened abdominal cavity after muscle repair, and this, combined with stretchy or lax tissues in some individuals, can lead to the result you are showing. So, thus far, it could be excess residual fat, or weak tissues, or both.
When you are laying down, does you abdomen flatten or become concave? If you keep your knees locked and do a straight-leg raise, what does your abdomen do? Protrude in the middle? Or just get firm and tight? These are helpful parts of determining an answer to the dilemma you pose, and this requires physical examination.
Mesh can help to reinforce a weak or stretchy abdominal wall, but remember that the mesh itself must be sewn to something. What happens when those tissues stretch and bulge? The fact that your breasts are showing the same characteristics may mean you are indeed one of those patients born with poor skin elasticity and collagen content, or an even more rare cases of genetically-inherited connective tissue disorder (such as Ehlers-Danlos syndrome) that causes a defect in collagen synthesis.
For breast revision and reinforcement, acellular dermal matrices (such as Strattice or Alloderm) can provide some additional support for breast lift patients who just have inadequate tissue of their own (very few, IMHO). These dermal matrices are very expensive, and of course still do not guarantee a satisfactory long-term result. Sometimes just a better surgical "design" or "pattern" that takes into account a certain degree of elasticity and stretch over time is a "better" solution, or at least the more common one!