Is it safe to keep reusing a portion of old dough? My grandma told my mum a weird process of making flatbread dough recently that she started to use. But I am not sure about it's safety at all.

Initially, normal wholewheat bread dough made with Instant Yeast. Used to make bread but a small piece is kept (~70-100gm) in refrigerator.
Next day, when making new dough, that 70-100gm portion of old dough is mixed in and it's allowed to rise for 2-3 hours.
A 70-100gm portion of this mixture is kept again in refrigerator and mixed in next day.
Repeat same everyday.

The result is definitely great for the flatbread naan but I am not sure about safety of this.
Question: Wouldn't a small part of it be too old after 2 weeks or so? I am sure some of of the old dough remains, however diluted. Mom claims she'd start process totally fresh after a month but I believe that's too long.
Edit: I heard it's similar to Pâte Fermentée but I'm still not sure if about a whole cycle.
This is how bread was made in my area just two generations before, not only flatbreads but every kind of dough. And the process wouldn't be started from scratch with new instant yeast, because it didn't exist yet.
I understand why you would say that a diluted piece of the old dough would remain after a month, but mind that fermentation is a metabolic process that converts molecules, and although actual atoms could be there, they were processed and rebuilt each time the dough ferments.
Compare that with any plant you eat; a part of this plant was a previous plant that was used as a fertilizer, maybe even passed through the animal metabolism. It's a cycle.