"I certainly can do good enough at manual testing but working purely on an automation project since many months, I am not confident about how well I will perform in a manual project."
That's true. The skills to explore a feature are very different than the coding skills to reproduce certain steps. To the point many people prefer to use the term "checking" to the latter and "testing" to the former.
"automation engineers can be appointed a manual or automation project with same level of efficiency expectations"
I have known many people that were very good in exploration and with coding for automation. Were they special/super intelligent? No, they just had the experience. People vary in their skills - it's a individual situation.
Do big companies keep their manual and automation testing team completely separate or is there an overlap between the two?
There are many which do, many which don't. Projects vary, because they are made by people. So, maybe keep different groups is more efficient. However, as a rule of thumb, small and integrated teams work better (Scrum is recommended for 8-15 people, e.g.), because it forces people to develop a T-Shaped skill-set.
Advice for your personal situation:
- Take the responsibility;
- Talk and pair with people who are more focused on your new role;
- Provide small chunks of results to your manager (reports and open places for exploratory charts, e.g.), ask for feedback (no more than weekly).