There are two different types of Big Bang models for the universe, and right now, astronomers cannot decide from their observations just what kind of a universe we live in. The first of these is the so-called 'Open and Infinite Universe'. In this universe, the Big Bang gave every scrap of matter in the universe more than enough speed that the gravitational force of all this matter acting on itself is not sufficient to slow the expansion down. The universe will expand indefinitely. In the second kind of universe the so-called 'Closed, Finite Universe', the Big Bang gave matter quite a lot of speed, but this speed is not sufficient for matter to expand forever. Eventually, the expansion will slow down, come to a stop, and then turn around into a collapsing universe. Neither of these two models look anything like a black hole because matter is ultimately free to travel as far as its energy will take it. The only limit we see is in the case of the Closed FInite Universe where matter is ultimately destined to recollapse to a 'Big Crunch'.
There is one sense, however, in which the universe does have a black hole-like property to it. All modern cosmological models predict that there is far more space in the universe than what we can see right now. If a light signal was beamed in our direction near the start of the Big Bang some 15 billion years ago, the farthest that light signal could travel since then is 15 billion light years. This means that, surrounding the Milky Way there is a spherical horizon whose radius is 15 billion light years that contains all the objects and points in the universe that we can now have seen since the Big Bang. BUT, if we travel to where a distant galaxy is located some 5 billion light years from us today, we can draw exactly the same 15 billion light year horizon around that galaxy. The only thing is that there will be galaxies that can be seen from this new vantage point that we cannot see in the Milky Way because the light hasn't gotten here yet. But it will get here in a few billion more years.
This information horizon is a feature of ALL cosmological models that have a starting point in time. In some ways, this horizon has some of the properties of a black hole's 'event horizon', but the physics is different. So, the answer to this question is that the Big Bang is not really similar to a black hole, even though both of these systems have an information horizon, but for different physical reasons.