This rule has been in effect for some time. From the park's own website:
https://www.parconazionale5terre.it/page.php?id=339
"IMPORTANT
OPEN FOOTWEAR AND/OR SMOOTH SOLES, I.E. NOT ANTI-SLIP (SUCH AS VIBRAM TYPE) ARE FORBIDDEN. THE TRANSGRESSORS WILL BE PUNISHED ACCORDING TO PARAGRAPH 2, ARTICLE 30, LAW 394/91 AND SS.MM.II
"On the trails it is compulsory to wear closed, waterproof and ankle covering shoes, with an anti-slip sole."
And there are also a number of recent articles out there on the subject, for example this from Fodor's:
https://www.fodors.com/world/europe/italy/cinque-terre/experiences/news/italian-officials-are-monitoring-tourist-footwear-in-this-region
"Officials in the park have also stepped up enforcement of footwear restrictions that were introduced in 2019. To hike the trails, closed, waterproof shoes that cover the ankles and have an anti-slip sole are required. Officials will be monitoring the trails for inappropriate footwear, and hikers found with footwear outside the requirements will be issued fines up to €2,500."
So even if not strictly checked in the past, there is the possibility that the Powers That Be will be looking more closely. As well, if you turn an ankle/ injure yourself badly enough to have to be rescued - no small feat given the terrain -due to inadequate footwear, pretty sure you'd be looking at a fine. I don't think you have to have boots, per se, but at least sturdy, closed-toe athletic shoes with a decent grip.
Having done most of the "Blue" myself, I sure wouldn't do it in just beach slippers anyway; too rocky, uneven and potentially slippery in places if wet.
Editing to add: I'm guessing you used the term "walking route" loosely but as a caution in case you haven't done the reading, the sections of the "Blue" between Monterosso and Vernazza, and Vernazza and Corniglia (the segment between between Corniglia and Manarola is closed) are not what I'd call a 'walk in the park.' Parts of them are steep and involve many rough steps.