Why don't we bleed interstitial fluid?
For fluid to flow from a wound there needs to be a significant pressure gradient between where it is now and the outside of the body. Your skin generally does not have a strong compressive effect, which is why a deep cut exposing fat will not lead to the fatty tissue being expulsed from the body any more than the interstitial fluid is.

Blood, however, flows. For it to circulate there needs to be a pressure gradient between where it is now and where it is going. Since veins (including the vena cava, which channels blood back into the heart) do not have vascular walls strong enough to create a suction effect (i.e. lower pressure than the surrounding tissue), you can conclude that the pressure of blood vessels is always higher than that of surrounding tissues, and thus higher than the pressure outside of your body. This is why all blood vessels, including veins, will bleed, whereas less pressurized systems such as interstitial fluid will not.