Reading through the administrative guidelines concerning the German nationality law (page 43 of this link), I found the following paragraphs concerning the loss of German citizenship when a foreign citizenship is applied for and awarded:
Ein Antrag im Sinne des Absatzes 1 ist jede freie Willensbetätigung,
die unmittelbar auf den Erwerb einer ausländischen Staatsangehörigkeit
gerichtet ist. Antrag in diesem Sinne ist damit neben einem
Einbürgerungsantrag auch der Erwerb einer ausländischen
Staatsangehörigkeit aufgrund einer Option, durch Registrierung oder
durch Erklärung.
Ergänzende Anmerkung: Bei einem Antrag ist zu unterscheiden zwischen
konstitutiver und deklaratorischer Wirkung. Bedarf es lediglich aus
formalen Gründen zur Inanspruchnahme einer ausländischen
Staatsangehörigkeit, auf die von Rechts wegen sonst ohne weiteres ein
Anspruch besteht, eines Antrags, einer Option, Registrierung,
Erklärung oder sonstigen Handlung, liegt somit kein Erwerbsantrag im
Sinne des § 25 Absatz 1 Satz 1 StAG vor.
With the following machine translation:
An application within the meaning of paragraph 1 is any free will
activity that is aimed directly at acquiring a foreign nationality. In
this sense, an application in this sense is, in addition to a
naturalization application, the acquisition of foreign citizenship
based on an option, by registration or by declaration.
Additional note: When making an application, a distinction must be
made between constitutive and declaratory effects. If an application,
an option, registration, declaration or other action is required only
for formal reasons to claim a foreign nationality, to which there is a
legal right otherwise, there is no application for acquisition within
the meaning of Section 25 (1) sentence 1 StAG before.
While the first paragraph seems to suggest that obtaining a foreign nationality (in this case Irish) by "registration, option, or declaration" would result in loss of German nationality, the second paragraph ("additional note") seems to walk this back, suggesting instead that when the "option, registration, or declaration" is "just a formality" needed to claim a nationality for which the person already has a legal right - German nationality is not lost.
My interpretation is that this "additional note" is intended to directly address situations like Irish foreign birth registration and clarify that such registration does not lead to a loss of German citizenship.