Most biographical information about Freud, starting with his date of birth, is falsified. Freud wasn't born on May 6, 1856, as the "legend" states, but two months earlier, on March 6. *  This is how Freud's true date of birth was found out.
When, in 1931, a committee of citizens of Freiberg ... set out to install a memorial tablet on the house in which Freud was born, it was discovered that contrary to all the “Who’s Whos” and to Freud s own statement, the city register lists Sigmund Freud’s birth on the day of March 6, 1856. (1)  
It was a marriage of convenience. Freud's mother, already two months pregnant, aged 20, married his "father", twice her age, on July 29, 1855. Subsequently, Freud was born 7 months and 8 days later, which makes him a "bastard".
Who was Freud's biological father? As Freud explained in the dream book, his eldest half-brother, Emmanuel, told him that it was he, who contributed his sperm to Freud's creation. Whether this information was correct, or a product of Freud's bizarre imagination, we will never know.
Remarkably, Freud's fake birth certificate, originally of a Christian child, is proudly being displayed on the Library of Congress' site.
* Bernfeld, Siegfrid. & Bernfeld, Suzanne, Freud's Early Childhood, 1944, p. 180-181.

Freud's birth certificate in the possession of the Library of Congress (LOC), even though Freud was Jewish, was issued by the Catholic Church.
The certificate is titled, Geburts- u. Taufschein which translates as a Birth and Baptism certificate. The Tauf part, thus baptism, as well as other printed words, referring to the Christian faith, were crossed out.
According to the certificate, the original record of the child's birth can be found in the tom (volume) VIII, p. 7 of the church registry of births.
Oddly, even though the certificate is for a Jewish child, the diocese, deanery, and parish are listed at the top right.
The form is bilingual. Some handwritten text is in German (Sechsundfünfzig [fifty six], Freiberg) and some in Slovakian (Maj, Pribor),

Circumcision

The information about circumcision, although not part of the orginal form, was added. Thus, the certificate states that the circumcision (beschnitten worden ist) was performed by Samson Frankel, from Ostrava.
As a matter of course, this kind of Jewish religious information would not have been recorded in the Catholic birth record or on a birth certificate.
According to the certificate, Freud was born on the 6th of May, and circumcised on the 13th of May in house number 117.
(For a Jewish boy, the circumcision would occur on the eighth day after the birth.)

 A combination of a record from the catholic birth register and a mosaic child is an obvious proof of its falsehood. Since Freud’s birth was registered in a Jewish registry, it wouldn’t be also registered in the church’s records.
Notably, the year, when the certificate was issued, was changed (overwritten) which confirms the suspicion that the certificate had been forged.

Which volume?

Notably, confirming the suspicion that Freud's birth certificate in the possession of LOC is a forgery, the tom (volume) VIII, specified on the certificate, starts in 1857 rather than in 1856!
And since Freud was born in 1856, it is apparent that the LOC's certificate was not issued for the baby Freud.
The only corresponding record on page 7 of volume VIII featuring house number 117, - the house where allegedly Freud was born - records the birth of a girl, Josefa, born on März (March) 5, 1857,
The girl was baptised, thus not Jewish. And since he was Jewish, Freud wasn't baptised, and thus his birth wouldn't be registered in the church books.

Which house?

It is not believable that Josefa’s mother would reside in the same house number, 117, as Freud’s. Thus, it is highly unlikely, that the Freud family, at the time of Freud's birth, lived there.
The only thing that Freud has in common with Josefa is the fact that both their mothers had children out of wedlock, the father unknown.
Without a doubt, even though the birthdate is off by a year, this is the correct record, since the book, page, and house numbers all match LOC's birth certificate.
Thus, the certificate was issued for a Christian girl rather than for a Jewish boy, and, subsequently, forged to appear to be Freud's.

This is the original record of Freud's birth as registered in the Jewish registry of births.
Since the word March in German is März, this being spelled with an "ä", it is apparent that the two dots over "a" are missing. On the other hand, the dots weren't missing in 1931, when the book was inspected by the committee of citizens of Freiberg, who then concluded that Freud was born in March.
Who, and when, falsified the record is not known, but it wasn't a difficult task. And the question of why it was falsified is easy to answer. At the time, it was a shame for an unmarried woman to give birth to a child.

According to the birth register of the Catholic Parish of Pribor (Freiberg) the only child born on May 6, 1856 - which allegedly was the day when Freud was born - was a Christian girl, Petronilla. Her record appears in the tom (volume) VII of 1856.
Thus, if Freud was born on that day, and registered in the church register, his birth record would appear next to Petronilla’s. This is not the case. 

Is there any corroborating evidence that Freud was born on March 6, 1856m rather than on May 6,1856? In fact, there is a confirmation, that Freud was born two months earlier than claimed, in one of Freud's high school reports. As it is apparent from the attached scans Freud's date of birth changed as he ages.
Thus, in grade one, according to his report card, Freud was born on März (March) 6, 1956.
(Notably, this record features the two dots over "a" missing in Freud's Jewish birth registry.)
Remarkably, already the next year, in grade two, at the stroke of a pen, Freud became suddenly two months younger, his birthdate stated as Mai 6, 1856.
This time also the month of May was spelled correctly as Mai rather than Maj, as on LOC's certificate.

The question of Freud's birthdate, especially during the height of his popularity, had been the question that many a Freudian scholar chewed over.
Thus, already in 1944, a Freudian scholar, Bernfeld, pointed out that Freud falsified the information about his date of birth. Again, in 1982, Balmary recounted the fact that, having seen photocopies of these registers, another Freud researcher, Granoff, found out that both Freud's birthdate and the date of his circumcision, in the Jewish registry, showed that Freud was born in April rather than in May. *

Qu'est-ce que c'est?

Nonetheless, the Freudian apologists, still to this day, ignoring the passage of time, claim that Freud was conceived appropriately on the wedding night, by his "father".  This is how Roudinesco, in 2017, argued that Freud was a "legitimate" child:
The birth certificate designates him by his Jewish first name, Schlomo (Shelomoh), born in Freiberg, on Tuesday, Rosh Chodesh Iyyar 5616 in the Jewish calendar, that is, May 6, 1856. The house where he was born was located at Schlossergasse 117.
Notably, one look at the LOC's birth record shows that this claim in several respects is wrong. LOC's birth certificate doesn't designate Freud by his Jewish first name, Schlomo (Shelomoh), but by his more gentile or "goyish" name, Sigismund. And the certificate doesn't specify his birthdate according to the Jewish calendar but according to the Gregorian calendar that has been in use since 1582.