Why do we as humans get embarrassed, some of us more easily than others and yet animals do not appear to have the same capability even though some are self aware?
Because human beings have the ability to see themselves through other people's eyes. Most animals do not seem to have this ability. Embarrassment is based on what you think other people think of you.
As humans we use it as a tool to gauge your standing in a social group. As social animals, being rejected by the social group can lead to death while rising in status can lead to better mating opportunities and access to resources, so shame and embarrassment are negative reinforcement tools to avoid losing social status.
Other mammals that live in social groups probably feel it or something similar.