What property of glucose makes it so that glucose found in (and is so integral to) so many disaccharides instead of a different monosaccharide, such as fructose or galactose, and why is it able to commonly form a bond with itself, unlike fructose or galactose?
Some readers seem absorbed with the notion that glucose can "form a bond with itself, unlike" other monosaccharides. This hasn't been addressed because it's an incorrect premise. To my knowledge this happens in many hemiacetals/hemiketals in water.
Glucose is a fundamental hexose; the metabolism of glycolysis is truly central to cellular life. This explains why it's so pervasive, in disaccharides or other carbohydrates. To use the wording of the question, the property of glucose that it explains its pervasive representation in disaccharides is its metabolic availability.
So why is glucose metabolism conserved across all three Domains of life? I explore two reasons.
Glucose metabolism could be central because of evolutionary history. Another type of aqueous biochemistry could prevail under terrestrial chemistries. So galactose-based terrestrial biochemistry is equally possible, in which case it would be overrepresented in biomolecules instead of glucose.
The aqueous abiotic (organic) chemistry of carbon (under terrestrial conditions) support glycolysis as a central metabolic pathway. So a galactose-based terrestrial biochemistry is thermodynamically disfavored.
(Same arguments go for pentose phosphate and TCA cycles, and the few other pathways that are so close to energetic equilibrium as to be amphibolic and metabolically central.)
Evolutionary biology provides evidence for explanation (1). This is because all domains of life share conserved enzymatic machinery for glycolysis; "The phylogenetic distribution of the enzymes of the glycolytic pathway confirms that it is indeed an ancient metabolic pathway, and was present (at least in part) when eukaryotes and
prokaryotes diverged about 1800 million years ago."From this reference, p. 224
There are also strong arguments for explanation (2). Below is one article detailing how glycolysis could very plausibly preceed cellular life. Glucose would then be pervasive, in part, because it's a pervasively metabolically-accessible (and chemically stable) sugar, under terrestrial conditions.