Ukrainian citizens who left Ukraine for Poland after February 24, 22 are covered by the Act on assistance to Ukrainian citizens. The fact of applying or not applying for a PESEL number does not change this fact.
Here you will find information on the rights and obligations associated with it: https://ukraina.interwencjaprawna.pl/the-act-on-assistance-for-ukrainian-citizens/
This means that the stay of such a person is automatically extended until March 4, 2024 and may be extended again at a later date.
However, if a citizen of Ukraine who came to Poland on February 24, 2022 or after that date does not have a stamp in the passport or other document confirming the date of arrival in Poland, he should apply for a PESEL number within 30 days of arrival in Poland – to prove the right to legal stay.
It should also be remembered that after the expiry of visa-free travel, it is not possible to legally travel to other countries (unless you apply for a visa or make sure that the country applies specific rules in the current situation).