Hi, I am interested in buying a 2013 Fiat Punto, however, it failed its MOT in both 2019 and 2020 due to the ABS light being on. Should this be something to worry about?

Yes..

Ask seller to pay for an MOT before sale..

You then know what is potentially going to cost

Very likely the original problem was never fixed if it appeared across multiple MOTs more likely a back handed had the light reset while the mot inspector looked at it, then it came back on again when the car was driven out.

Lots of shoddy things happen with dash lights for MOT. Many years ago I got taught how to fudge an oil pressure light if you just want to sell a car, or in the case of a taxi get it through whatever the taxi version of an MOT is.