The simple descriptive equation for a black hole relates its radius, R, to its mass, M. R = 2GM/c**2 where G is the universal constant of gravity, and c is the speed of light (here squared). it is interesting to note that that same equation can be derived using both Newtonian Gravitational Theory and Einstein's Theory of General Relativity.
Within the range of uncertainties in our knowledge of the radius of the observable Universe and the mass contained within the Universe, that equation is satisfied even now, some 13 billion years after the Big Bang. In that sense, the Universe does have the apparent characteristics of a black hole, although that might also be nothinng more than an interesting coincidence.
A black hole, whether a large one at the centre of a galaxy, or a smaller one which resulted from the evolution and death of a massive star, is an object/structure/thing that exists within space and time. That is, a 'normal' black hole is an object existing within the much larger Universe.
The Big Bang event was not an explosion which occurred within a large, empty space. Rather, the Big Bang was the initial unfolding of space and time. All parts of the Universe were participants in the Big Bang event.
When we talk about the expansion of the Universe as revealed by the recession of galaxies, it is space which is expanding and carrying the galaxies along with it, and not the galaxies which are 'really' moving through space. Hence, there could in principle be galaxies beyond the edge of the observable Universe which are apparently receding at a speed in excess of the speed of light. We could not see them, of course, and because they are not traveling through space faster than c the fundamental principles of Special Relativity are not violated.
By the same token, at or immediately after the Big Bang, all the matter in the Universe was tightly confined but so too was space and time. The scale of the Universe in all respects differed from what it is now, and there was no 'outside'. An hypothetical observer inside -- as all observers must be unless we live in a higher dimensional hyperverse -- would see an unfolding of space and time as we do, with matter carried along in the spacetime continuum, and would observe nothing that contradicts the principles of Relativity.
I might also say that a black hole is defined as a region from which no signals can escape to infinity (meaning to infinitely far away, assuming that there is such a region). If our universe expands forever, and if we are in a part that also expands forever, then we are not in a black hole. (The Big Bang need not be inside a black hole, contrary to some popular thinking that is reflected in the original question.)
If the universe does not last forever but recollapses, then effectively the entire universe is inside something very much like a black hole. Then we have not escaped, but we and the rest of the universe can last for a long but finite time.
Technically, if the universe is roughly the same everywhere and recollapses eventually, there simply is no infinitely far away region where light rays can continue on forever, and so the definition of a black hole would not apply. So even then we would not really be in a black hole, since then there would be no outside region that could not receive signals from us. Of course, if our part of the universe recollapses, for us it is effectively the same whether or not there is an outside region that survives, but I'm just saying that by the technical definition of a black hole, this scenario also would not count as having a black hole.
One could be even more pedantic and note that if the whole universe recollapses, even stars that collapse much sooner do not really form black holes, since there is no outside that lasts forever. But if the universe as a whole lasts much longer than the region that collapses inside a star, we bend the strict definition of a black hole a little bit and say that the region that collapses is a black hole, since it cannot send signals out, even if any other signals that are outside do not really go to infinity. In this case we say that signals outside can go a long way, and for practical purposes this is almost the same as saying they can go forever, so if signals inside the collapsing cannot go nearly so far, we physicists generally say they are inside a black hole.
If you proceed in this way of bending the strict definition of a black hole, to bend it not just a little but a lot, you could similarly say that everything inside a universe that recollapses is also inside a black hole, even though in this case there are nothing outside that lasts any longer than the inside. Physicists don't usually go so far in bending the strict definition of a black hole, but that is a matter of taste and of definition.
The main point is that there are regions inside collapsing stars that can't have any signals survive very long, and the same might be true (but need not, if the universe lasts forever) of the entire universe (where very long might mean very, very long). Physicists usually use the word black hole for the former case but not the latter, but if the whole universe recollapses, it is a quantitive difference rather than a qualitative one, so it wouldn't be too bad an error in lay discussions to say that then the whole universe is inside a black hole (so long as one does not mislead anyone into believing that in this case there is an outside to the black hole, as there is for what physicists usually call black holes).