Any idea what age the circuit is? It looks like 1970s to me. Our intercom is of a similar vintage and is quite crude - the microphone is a carbon one.
Form your rather grainy photo, I'd say the buzzer is maybe the thing that looks like a relay in the top center. It has two wires coming out of it. One goes to the green terminal block and the other goes to the white terminal block. They're supposed to be terminals "5" and "7". I'd put a voltmeter across those two terminals and get a friend to press the downstairs button. The buzzer is probably AC so you'll need to set the meter accordingly.
Is the buzzer 6V or 12V? It depends on the model number.
If it's AC, you'll need a rectifier in order to operate you extension buzzer. I wouldn't use a relay, I'd just use a modern electronic sounder. They take a very small current compared with the ancient buzzer in your intercom. Look on eBay for "sounder 6V" or "sounder 12V". The first hits I got are:
https://www.ebay.co.uk/itm/372371793332
https://www.ebay.co.uk/itm/125387551594
Also search for a "bridge rectifier", e.g.
https://www.ebay.co.uk/itm/285022072920
(you may want to limit the search to your own country so as to get quicker delivery.)
And a smoothing capacitor - e.g. 470uF 25V
https://www.ebay.co.uk/itm/202799914226
Do you need help with the circuit diagram?
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Here it is. Note that each component should be marked with "+" and "-".
