In this context it is a synonym for jus soli - birth happened on the soil of the country.~
Birthright has a few other meanings in wider contexts: https://www.collinsdictionary.com/dictionary/english/birthri...
Does it really matter if we add one more, especially as it is now a well established usage?
Yes it matters, because jus sanguinis is also a birthright and therefore "birthright citizenship" despite having no relevance to where you were born (jus soli)