Kortuk, our beloved mod, showed me that if someone clicks to my name in chat, at the field
Parent user there is the link to my tex.stackexchange, instead of electronics.* one.
Is this a bug? Is it just the behavior of the chat?
Each chat user is associated with a parent site. This allows the system to determine things like whether you're allowed to chat (does the parent user have 20 rep?), whether you can create rooms (does the parent user have 1000 rep?) and whether you can see flags and enter private rooms (is the parent user a moderator?). These privileges are on a network-wide basis; it's not segregated by sites.
If you want to change the user to your Electrical Engineering account, go to your chat user profile (click your avatar in chat and pick 'user profile', which would take clabacchio to http://chat.stackexchange.com/users/30500/clabacchio), and click 'change' next to the parent user link:
(Note: Non-mod and non-clabacchio visitors to that page will not see that link)
This will bring up a dialog where you can type the site name that you'd like to have associated with your chat user. Note, though, that if you change your parent user to your Electrical Engineering user, it will still point to the Electronics user when you're in the Tex, LaTeX, and Friends chatroom.
This is not a bug, but I can see how you might be confused.