The bulb itself looks like a standard W5W bulb, but has two separate tungsten wires, so needs 3 or 4 connectors (ground, possibly another ground, +12V 5W, +12V 21W).
Another question is how is it mounted in the light cluster. Can't find any photos online

Most likely it is a specific mount, but also could be something common.
Looking at those threads:
Hi guys newbie here but with 2 daughters both with Fiat 500’s may be spending some time here lol I’m just changing the daytime running light bulb on 2013 model - bad enough having to take wheel off to get at it but can anyone suggest a source for new bulb holder - this one seems to have...
www.fiatforum.com
The day running light connector has corroded on my 2012 500 1200 and I cannot find a replacement, any advice?
www.fiatforum.com
it seems that the problem is common, but there was no solution though.
I can't find the appropriate wirings on
ePER. Pity.
I had a very similar problem on my girlfriend's non-FIAT car rear light last week. The bulb holder contacts corroded and one of them broke, so the light was not working, I had to cut the wire and solder it directly to the bulb. The car is 18 years old, the wiring is no longer available at the dealership, I've found online a guy who's selling this used part for 100 EUR and new ones for 170 EUR. Plus P&P.
But having found the offer with photos, I've used google lens and found that 5 or 6 another car models use exactly same holders! And a part of them is very popular. Of course the plug that goes to the loom is different, but the problem is in the bulb holders. I've bought the whole light cluster with the loom for around 15 EUR delivered and will resell the cluster

Then I will just cut the holders and replace in the loom on the car.
Try finding the photo of the wiring needed with the broken holder, or take it off the car and take the photo yourself and then use google lens to find the whole wiring loom. Should be much easier than finding the holder itself.