Groups are not generally a widely used thing in issues, so there's not a huge amount of functionality around them. But they definitely can have a lot of uses for some of us.
Rather than my usual rambling block of essay, I'll try to focus on your questions for a change.
But before that, I want to make it clear that it is very unlikely that your people are "singletons" - they can, and almost all will, belong to several groups. This can make seemingly simple requests quite complex. A common question in the community is something like "how do I make issues visible only to people in the same group as the assignee?" - the answer there is technically a simple "do nothing", but it's because all the users are in a group that says "this person can log in", so everyone is in "the same group as the assignee". However you read the below, you need to bear in mind people can be in many groups, and it's very likely that everyone is in a single "can log in" group at a minimum.
>Is there a way to see this groups inside an issue?
Not off-the-shelf. Jira issues focus on the people involved, not the groups those people are in. You will need to play some tricks (see below)
>Can I map it to a field?
Sorry, same as the question above
>Or at least can I search and filter by groups?
I have asked for more, but you can do a basic search with the "membersof" JQL function.
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