What it means is that they don’t know what they’re doing.
There’s no such thing as “Full ESI”. There used to be API keys, which came in two flavors (limited and full). Then that was expanded to allow for more options. (Bad corporations asked for everything. Some ok ones did too, but it was indicative of people not caring.)
Now there’s ESI, which is an API that gets used by people logging into websites, in a similar way to when you use twitter or facebook to log into a site. You never give your username and password to anyone but CCP, but you delegate some rights to access information about you to whoever is running that site (There’s a list which gives some basic details)
So if they’re asking for you to give them something, they don’t know what they’re talking about, because there’s nothing to give. If they’re asking you to log into a site, it’ll tell you what (in general terms) they’re getting.
They should not be asking for it all, and if they are, you should name and shame them, as people who don’t have a clue. I can see no good reason, for example, for a corp to be able to send email as you.
I wouldn’t be too surprised if they’re asking for:
Wallet details (to see if you’re a spy)
Asset Details (to see if you’re a corp thief)
and so on.