I presume this might have also been transliterated as Reichla. When being registered somewhere, either the person just resigned themselves to being referred to as Rachel, or some other registrant just made it so.
This is a similar problem that I had. I had gggrandmother whose name was pronounced approximately as Reichel, or sometimes Rechel. But this was different from the Biblical Rachel. As proof, in Slovakia, my father's eldest sister was a Roze, with Hebrew name Rachel. Perfectly correct.
A younger sister, with a civil name of Edith, had a nickname or "Rechi" and her birth record clearly had her civil name Edith and underneath the Hebrew: רעכל. A snip from her birth record is attached. This name is derived from a German name and probably related to "Reich" meaning rich. I may have see it also meaning something like "blessed." (Maybe if you are rich, you are blessed?) And meanings of the name Edith, can be riches, or blessed.
In the birth record, you may also see that my grandfather was a learned man, and had indeed attended the Pressburg Yeshiva. So he would have known the difference in the names and would not have given two daughters the same name.
Additionally, the spelling רעכל is definitely different from רחל. On tombstones that I have read, I have seen other curious spellings such as ראכל and רעכיל. Those of us who have arranged for such stones for family members know that each letter added to a marker adds to the cost. So if her name had truly been just the three letters of רחל, they would have saved some money.
All that being said, this variation of the name has lead to later naming issues in family's trying to honor their ancestors but not knowing the difference!
Beverly Hills, CA
Researching: GOLDMAN, STEINER, LANGER, GLÜCKSMAN, STOTTER in various parts of Galicia, Poland
(Nowy Targ, Nowy Sanz, Wachsmund, Dembno, Lapuszna, Krakow, Ochotnica) who migrated into Kezmarok or
nearby towns in northern Slovakia and Czech Republic (i.e., those who lived/had businesses in Moravska Ostrava);
GOLDSTEIN in Sena or Szina, Szkaros and Kosice, Slovakia; Tolcsva and Tokaj, Hungary.
GOLDBERG, TARNOWSKI in Chmielnik and KHANISHKEVITCH in Kielce, Poland