Sounds like you are trying to measure in parallel with the positive lead wire, which is what you would do if you were trying to measure the voltage drop across the wire rather than the current running through it.
To measure current, you must place your leads in series with the wire, so you must disconnect either your positive lead or your negative lead and put your test leads in line with the circuit.
CAUTION: Your LED driver can output up to 118Vdc, which enough to cause SERIOUS harm. DO NOT TOUCH WHEN LIVE
So, in your circuit, before powering up, disconnect the “+” wire coming from the LED driver and connect the “+” lead from your ammeter to the LED driver. Then connect the “-” lead from your ammeter to the “+” input wire which goes to the “+” Wago connector. Make sure your ammeter is set to the right scale and that the leads are connected correctly on your meter (most multi-meters use different connections when measuring current rather than voltage). Once you have double-checked your connections, and ensuring that you are not touching any live wires or contacts, you can apply power and measure the current.
If interested, you can also measure the current in each of the three branches by following the same procedure. Just disconnect the lead going to any one of the three branches and put your ammeter leads in series there. The currents should be very similar between the three branches, but they may not be identical, due to variances in characteristics of each LED strip.