Infinity has different meaning in different context. For example, in real analysis, infinity is considered as a “symbol” means $+\infty$ is a symbol and $-\infty$ is a symbol and there is no notion of equality between symbols, on the other side, at some places, infinity is considered as a number. In set theory, we study about infinite sets and here there is no notion of positive infinite sets or negative infinite sets. And there might be other places where infinity has different meaning. So, to understand whether $+ \infty$ is same as $-\infty,$ we have to understand the context, we have to understand the meaning of infinity.
In probability theory, some people say probability is always positive but it is not true. Probability can be negative. I can show it in a very good book where notion of negative probability is given. We say positive probability because we follow the theory which is based on the assumption that probability has a value between 0 and 1 (including both). If we break this assumption then probability can be negative too and it might be possible on some other planet, there is a notion of negative probability.