I bought a single ticket from Lufthansa, of Mumbai to Paris. My first flight (LH765) lands at 5:55am at Munich, from where I have another Lufthansa flight (LH2226) to Paris at 6:40am. This is a single ticket so I guess its responsibility of the airline to get me on the second flight, but since it's Non-Schengen to Schengen, will customs etc. be a problem? I am assuming the checked-in luggage would transferred automatically by the airline. Will I be able to make it?
First, your baggage will be checked through, and if you miss the connection, Lufthansa will put you on a later flight at no extra cost. Since you arrive early and there are many flights between Munich and Paris, it will probably be only an hour or two later.
You will have to go through immigration and possibly security, but at 6 AM there probably won't be any queues. It's a large airport though, and walking time alone will be significant.
All in all, 45 minutes is doable, but there is no safety margin, especially if your first flight is delayed.
See if you can get a seat at the front of the plane so that you can get out quickly. Ask the staff about it when you check in.
You will arrive and depart in Terminal 2 which was specifically designed for short transfer times down to 30 min. There are even extra cars which take you directly from plane to plane (via security) if needed, see: http://www.munich-airport.de/en/consumer/aufenthalt_trans/airportstop/minconntime/hcc/index.jsp
Most likely you will arrive in Terminal 2, floor 5 or 6 (international) and need to get down to floor 4 (Schengen), passing EU/Schengen immigration on the way. If you get bussed to the terminal, you arrive in floor 3, go through immigration and then upstairs.
Baggage will get checked through, customs will be only in Paris. If your flight is delayed, tell staff about your connection, they might be able to organise something for you or have additional advise. You might still make it but perhaps with your baggage delayed. Terminal 2 is long (1000m) so some walking involved, but there are Moving Walkways.
If the booking system offered the connection, it is doable. Also, if I remember correctly, LH is using the same terminal for all their flights, so you would not have to change terminal.
The connection is tight, and there are chances that you won't make it. You will then be automatically booked on the next available flight. If the baggage makes it but you don't it will be held for you at your destination.
If you will see that the connection gets really tight, you may notify the flight attendants, who have procedures. If the connection is still doable, there may be ground staff waiting for you (and fellow passengers to that flight) who will guide you to the departing gates (getting you through immigration and security faster).
I have not much changed planes in München, but in Frankfurt, which is bigger, and there were several 45 minute connections working without problems.
You'll manage it, otherwise Lufthansa wouldn't have offered the connection.
Since yours is a long-flight from Mumbai to Munich, there is a chance that you come earlier, and in reality you have 1 h.
The only caveat is that from my personal experience, your checked-in luggage possibly won't manage to catch the next plane. Missing baggage after connected flights was pretty common for me. Maybe since it's the same company, then, you'll have a better chance, but even then, think about what you won't get in the first few days in Paris.