The standard method to measure temperature using a diode, is to measure the voltage at two (or three) different currents. If you use the LED, then you are measuring the junction temperature.
For example at currents of 50mA and 5mA.
The difference in the diode voltage at these two currents is a fundamental function of the diode curve, and the ratio of currents (e,g. 10x), and is quite interchangeable between devices of the same type. (whereas the simple voltage-temperature curve is not).
The trick with LED's is switching the currents quickly from the running current (which causes heating) to the measurement currents, fast enough that junction temperature is unchanged. You can use capacitors to sample and hold the voltage difference quickly, and measure this more slowly with an ADC.
simulate this circuit – Schematic created using CircuitLab
From memory, the system cycled RunCurrent(1ms) I1(20us) RunCurrent(1ms) I2(20uS). SW1,2 are analog switches or sample and hold amplifiers. The fet gate drive invertors are running from 12V. The measurement currents are switched into D1,2 when not used. This allows continuous current flow in the constant current sources, so giving fast accurate currents compared to turning the constant current generators on and off.
Calibration can be done with runcurrent=0, so you measure temp of water bath or hot air.
The schematic was for a single led measuring arrangement. If I was doing it for a led strip, and needing to individually measure all leds, then the current switches would be arranged to control the whole led string, and the led voltage sampling would be separate and isolated, so you can probe across the individual leds.
TonyEE can probably comment on the usable current range where modern GAN leds behave as diodes. I have only used this for older types of heterojunction leds.
Extending this to using 3 currents eliminates resistances, but LEDs are known to behave peculiarly at low currents, so this might not work.
Some leds have a reverse protection diode, and this could be used as a package temperature sensor instead.
A variation on this principle would be to vary the run current by a small amount, e.g. 10% i.e. if LEDs are normally running at 200mA, then switching them 190-210mA @ 50% duty cycle. Now the change in voltage across the leds is very small, but it is a simple AC signal that can be measured with a good meter or amplified easily. The drawback of this method, is that calibration will be difficult, as it is being done at full led current.