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How to identify the the shape of probability distribution?
You cannot. There are hundreds of different distributions. The shapes of the distributions depend on their parameters so that the same distribution can be symmetric when the parameters have some specific value, but is highly skewed - in either direction - for other values.
A distribution that has the same shape on either sife of the center is said to be symmetrical?
true - Distributions that have the same shape on both sides of the center are called symmetric
When two distributions have the same mean but different distribution and variability?
The answer will depend on what the question actually is!
When it comes to comparing data from different distributions what is the benefit of normal standard distribution?
There may or may not be a benefit: it depends on the underlying distributions. Using the standard normal distribution, whatever the circumstances is naive and irresponsible. Also, it depends on what parameter you are testing for. For comparing whether or not two distributions are the same, tests such as the Kolmogorov-Smirnov test or the Chi-Square goodness of fit test are often better. For testing the equality of variance, an F-test may be better.
What is the same shape but different size?
The word for this is "similar." The same shape and the same size is "congruent."
