Plantagenet was a name assumed by Richard, duke of York in the fifteenth-century for his 'lineage'. It was a call back to a sobriquet (nickname) of Geoffrey's but no king between the twelfth and mid fifteenth-centuries considered themselves a 'Plantagenet' ruler.

In reality 'family' names were not all that common until the fourteenth-century and even then were not essential. Thus you would be identified as Henry of Bolingbroke (ie. either 'from' Bolingbroke or 'lord of' Bolingbroke). This might mean you had a couple of these types of name, or that your name might change over time (Henry of England). This is the pattern almost everywhere in Christian Europe (excluding patronymic regions such as Wales, Ireland, and possibly Scandinavia). The Lancasters and the Yorks are identified because that is where their ducal titles heralded from, they also had a close affinity with the regions (unsurprisingly).

Another note:

The term 'House' is also an anachronism in the Middle Ages. The word for 'household' in medieval Latin was familia - I'm sure the modern English derivation is familiar (pun intended). So when historians discuss House of [x] and [y] they are simplifying the past for their audience.