I've arrived in Paris without ID, nobody checked my passport through whole process

I was flying to Bogota from Manchester with stop over in Paris, I've left my passport by security in Manchester airport and nobody checked it by the gate and entering plane, guy on the gate also let me to the plane with 20 kg bag as I was running late, I've spent about 700 pounds extra to get new flight to Colombia, I went through big embarrassment and lost 2 days of my holidays, what recompensation I should get from Flybe?

I'm not sure I understand your argument. Are you saying that since you misplaced your passport, you should not have been allowed to board (because they should check such things at the gate). Had you been denied boarding, you would not have run into problems at the destination?

I know in some airports, it is common to look at both the passport and the boarding pass; but it does not have to be a passport. The airline can ask for any form of government ID that is accepted for identification purposes.

For example, citizens of GCC countries can travel among GCC member states simply by showing their national identification card - no passport is required.

I didn't show my passport to anyone at the airport, anyone...that means anyone using my boarding pass could enter the plane. I had a big 20 kg bag with me as hand luggage too as I was running late and was expected to pay but they didn't realize with this too. I was denied boarding in Paris to they deport me back to Manchester and I've been told I shouldn't be on this flight and airline will be charged 5000 euros for everything. I know it was my fault leaving my passport by security but airline is responsible to check my ID to make sure its me.

Checking the passport/documentation/id is the responsibility of:

1. The airline when checking you in. They need to ensure you have the proper documentation as the airline is normally fined (as you found out) if they carry passengers are that not eligible for entry at the destination. In fact, some airlines specifically have this disclaimer (which you must agree to) that they are not liable for any refunds if you are deemed ineligible at the port of entry.

2. The immigration officers at the airport (if applicable - for example, there are no exit formalities in the US). These officers are the sole authoritative confirmation of your immigration status/documents. They sometimes check your documents on departure, they always check your documents on entry - even if you are a citizen of the destination country or are otherwise similarly exempt from any visa/documentation requirements.

Throughout the rest of the journey - you are only responsible to carry government issued identification - and it does not need to be a passport. A resident card, or sometimes, even a driver's license is sufficient.

I flew from Amsterdam - Geneva - Malaga and no one asked for my passport. I only had to scan it at the kiosk to print my boarding passes.

Boarding passes once scanned for entry, cannot be used again.

Lets say someone made a copy of your boarding pass. They approach the gate before you, scan the boarding pass and board the aircraft. Now you arrive and try to scan your boarding pass - the gate will not open (in case its an automatic gate); as your boarding pass has already been utilized.

At this point, you will be asked to provide some identification.

I've paid another 500 pounds to get another new flight to Colombia plus internal flights

It it unfortunate that you were not stopped at your flight from Manchester to France and you had realized you had misplaced your passport. However, as you admit it was your fault, I am not sure what compensation you can expect from the airline.