Well, my ESP8266 talks with the MQTT broker (mosquitto), as the log shows:
1557439140: New connection from 192.168.1.5 on port 1883.
1557439140: New client connected from 192.168.1.5 as Lamp-0101_1 (c1, k10, u'').
1557439593: Client Lamp-0101_1 disconnected.
1557439652: New connection from 192.168.1.5 on port 1883.
1557439652: New client connected from 192.168.1.5 as Lamp-0101_1 (c1, k10, u'').
1557440104: Client mqtt-explorer-022602ca disconnected.
1557440136: Client Lamp-0101_1 disconnected.
1557440143: New connection from 192.168.1.5 on port 1883.
1557440143: New client connected from 192.168.1.5 as Lamp-0101_1 (c1, k10, u'').
1557440236: Client Lamp-0101_1 disconnected.
1557440247: New connection from 192.168.1.5 on port 1883.
1557440247: New client connected from 192.168.1.5 as Lamp-0101_1 (c1, k10, u'').
The firmware on this ESP8266 is ESPEasy.
Any link on how to set it up manaully, as a generic mqtt thing?
You will have to set it up when programming your ESP8266. It depends on where you have required the code from but most of them have section somewhere that you need to define the wireless SSID, password, device or MQTT ID and inbound outbound channels.
A client (your esp device) needs to define these details when sending a message.
FYI. I have been using MQTT toolbox or MQTT FX (not at my pc ATM) WHICH IS super handy to test channels and confirm the talk on these channels.