No. A block is not armour. It provides a minimum level of defense/resistance, but does not stack with 'active' defenses such as subsequent opposed rolls.
I agree with your second and third sentences, but fail to see how they are in any way relevant - neither armor nor active defenses are involved. A block, very simply, prevents actions with fewer total shifts, or reduces (by the block strength) the effective shifts of an action that beats the block.
For example: You use a gun to lay down covering fire, establishing a four shift block against people trying to move out of some zone. I decide to take a sprint action, and get five shifts on my athletics check. End result: I get to move one zone (spending most of my effort vaulting over your line of fire), not five. This does not break your block, however, and anyone else trying to move still needs to contend with there being a hail of bullets in the way.
Example Two: You grapple me, establishing a four shift block against everything. I decide to attack you, and get five shifts on my fists roll. Because this is an attack, you also get to roll defense - say you get a three on fists to defend. Those three shifts don't stack with the block from your grapple - you take just the better result for your defense, which in this case is the four shift block. I hit for one shift (plus any weapon rating I might have from, say, Claws, and minus any armor you might have). This also breaks the grapple.
Example Three: You grapple me, establishing a four shift block against everything. I decide to counter-grapple, and get five shifts on my might roll. Because this is not an attack, there is no active defense - but my effort is still reduced by your block, resulting in a one-shift grapple on you. This also breaks your grapple.