In origin, harbour and auberge are the same word. Both occur as English terms in the OED, which gives the word in Anglo-Saxon to be herebeorg, becoming the modern German herberge. Its original meaning is shelter for an army, but in English it was soon used more generally, to mean any shelter or lodging or the place of it. Its use for ships is metaphorical, not the other way around.
To harbour someone is literally to give them lodging: this is the primary meaning of that word and still its meaning in law. In one notorious murder case well within living memory, the police knew that the murderer must have been assisted by his girlfriend, but as yet had no evidence to charge her. However, after the crime he had slept at her house; so in order to take her into custody, they used an old formula, and charged her initially with the crime of having knowingly comforted and harboured a felon*: i.e. she had given him food, drink and a bed for the night.
A harbinger originally means a person who goes ahead of an army seeking accommodation for it and announcing its arrival; then more generally it came to mean any person, or even thing, that announces an impending arrival. Useful, because there is no other word; and it has taken the fancy of poets because of its attractive sound.
The words in European languages, both now and in ancient times, for inns, guesthouses, hotels, taverns, roadhouses, lodging houses, dosshouses, boarding houses, B&Bs, etc., each with its own flavour, are bewildering—as is, in our own time, the spectrum of expectation that people have of all such places, confirmed by the extremes of contradiction to be found on a website like Tripadvisor.
So what kind of place was the inn where Mary and Joseph wished to stay, and what is the Greek word for it, as used in the relevant passage of St Luke’s gospel?
…καὶ ἔτεκεν τὸν υἱὸν αὐτῆς τὸν πρωτότοκον· καὶ ἐσπαργάνωσεν αὐτὸν καὶ ἀνέκλινεν αὐτὸν ἐν φάτνῃ, διότι οὐκ ἦν αὐτοῖς τόπος ἐν τῷ καταλύματι.
κατάλυμα is literally an unyokery— somewhere where you could unload the animal you were leading—whether ass, mule, camel or, more rarely, horse—and get shelter for it and yourself. We can assume that it was pretty basic.
* This would not have been an offence if she had been his wife.