My understanding is that firstborn is a “legal” term to designate the first child born, regardless of whether or not there are any subsequent children. An only child would be designated as “firstborn” as well as any eldest child.
You are partially correct. First born doesn't always mean there are others.
However, if we examine the Greek some light is shed on this topic.
The Greek word used in Luke's account to describe Mary's first born is πρωτότοκον (πρωτότοκος). From HELPS Word-studies it means first, preeminent. Strongs Concordance notes it to mean first born, eldest.
It is the word we derive prototype from. A prototype is the first...usually with more to follow.
It allows for others to follow.
If Luke had wanted to say this was Mary's only child he would have used Υἱὸν τὸν μονογενῆ. This is what John used in John 3:16 in describing Jesus as the only Son of God.
The Greek μονογενῆ means one and only; one of a kind. Υἱὸν means son. Hence, the only Son. We know Jesus is the only Son of God.
That Luke did not use this greek word choice tells us that the door is open for Mary and Joseph to have other children. As we examine the text of the NT we find they did have other children when we see references to the brothers and sisters of Christ in the NT.
Hope this helps.