Now who is responsible for the missed flight?
Unfortunately, you are there.
When flights are purchased as separate tickets, the responsibility of the airlines does not flow – their obligation to you ends when you reach your destination on the ticket. In this case, you had two tickets with two destinations. This basically defines everything here.
The second plane left at the time and your boarding was not denied. So, in the eyes of the airline and, above all, the EU regulations on compensation, you simply have not managed to introduce yourself.
The first plane took you to your destination, your connecting airport, 90 minutes late, but according to your itinerary on this ticket, the connecting airport was actually your "final" destination for this ticket; to obtain compensation, you must have a delay greater than 3 hours at your final destination.
The EU261 FAQ has this to say about missed flights due to delayed inbound connecting flights:
Do I have any rights if I am not allowed to ride in my connecting flight because I arrived late at the gates because of a delay with the first flight?
YES – if the flights are part of a single booking, the carrier must offer you the option between the refund of your ticket and a round trip flight to the departure airport as soon as possible or rerouting to your final destination at the earliest opportunity for re-routing at a later date at your convenience under comparable transport conditions, subject to availability of seats. If you are rerouted and you reach your final destination with a delay of 3 hours or more, you are entitled to compensation.
Note the very specific mention of a single booking. Your tickets are multiple and separate bookings.
You can contact the airline to find out if it can book you a cheaper flight or no cost, but it has no obligation in this case.