Birthplace in the document only shows: Egyhazas, Hungary
There are hundreds (or more) place names in Hungary with variations of Egyhazas.
How was this particular one selected?
Link to the indexed record
"How was this particular one selected?" Randomly, by a bot. It's probably what came up first in a database lookup based on the text.
Unfortunately, due to the widespread depredations of the autostandardization bot, nothing that FamilySearch's database says about "where" can be trusted any more. Even more unfortunately, FS has so far failed to acknowledge this fact, and continues to expect to fix the errors piecemeal, based on individual user reports here in Community. And this totally leaves aside the uncorrectable errors, like this one: there is no provision for leaving an indexed place field unassociated with a location in the places database.
(The 1913 gazetteer lists 17 places in Egyházas-, along with 19 places in -egyháza, 11 in -egyház, and 14 in Egyház- [literally "one house": "church"]. Depending on the date of that naturalization, I would have a starting guess of Kis- or Nagyegyházas in Nyitra county, but these would be highly uncertain possibilities, given that they were in different districts and originally called Kosztolnafalu and Kosztolna, respectively.)
Ilona's arrival: https://www.familysearch.org/ark:/61903/1:1:JXLW-1J4
...which doesn't help. The current gazetteer lists an Egyházaskozár in Baranya county (https://www.ksh.hu/apps/hntr.telepules?p_lang=HU&p_id=27401), but in 1907, that was called Ráczkozár, and in any case, the entry looks more like Egyházaskaszaj or thereabouts to me -- and I'm pretty sure that it's followed by "Vas" in the departure contact column. Vas county had seven places called Egyházas-something (Egyházasfüzes, Egyházashetye, Egyházashollós, Egyházaskesző, Egyházas-Nyck, Egyházasradócz, Egyházasszecsőd), but none of them are anything much like "kaszaj"; the closest is probably Egyházaskesző. And her birthplace column (on the next image) again unhelpfully truncates it to just "Egyhazas". Ugh.
For whatever it's worth, it looks like the naturalization index is only showing the (very likely completely wrong) place in Pozsony county in the search results display, i.e. it's an on-the-fly autostandard that's only used for the filters, and doesn't actually appear in the individual index entry.