In some Municipalities in Poland, are introduced special web-platforms, which provides access to wide range of City Offices, you can send some drafts or documents via net, you don't need to go by foot to some places, just send it, and you'll receive answers and results, sooner. PESEL is usually your nick or password.
Next, e.g. at uni or in library, PESEL is used as your Password, look at the WKLBN - you have there the biggest libraries in Lublin, i.e. Maria Curie Skłodowska University, Catholic University of Lublin, Municipal and Regional Library, Polytechnics, Medical University, Agricultural University and others...
Thanks to PESEL, students have those services centralised.
I know the government is thinking of getting rid of them
You're wrong, they want to upgrade it to the PESEL2... because PESEL (ver. 1) has some technological restrictions...
Probably, you thought about "Meldunek" (Propiska) - those are informations about place of residence...it differs to PESEL, Meldunek is a Communist relict, and Polish rulers want to import legal solutions similar (the same) to those in UK...
Sometimes, I think, they forcefully want to make Poland, the Second UK...