The text of the Teleport spell strongly alludes to the “familiarity” aspect of the spell being related to how uniquely identifiable a destination is starting with this “You must have some clear idea of the location and layout of the destination. The clearer your mental image, the more likely the teleportation works.”
The more a location is unique the clearer the anchor points in your mental image are likely to be.
None of the wording in the categories of familiarization mention having to have stayed somewhere a long time. Frequency of visits gets a mention in ‘very familiar’, though I suspect this is just the most obvious way a person can know a place well.
Imagine teleport destinations as needing spatial co-ordinates, but instead of numbers it uses unique aspects of the locale. For instance a real world locale would use longitude, lattitude, and height above sea level, in teleport terms you would need a similar number of unique reference points to get a “very familiar” destination.
For example, the shingle above the inn door has a green dragon curled around a brown tankard and the words ‘Drunken Dragon’ in Common on it, the cellar door of that inn is at the bottom of steep stairs and is painted red with a small face hatch in it, there is a hangling oil lamp light right over the cellar door which has distorted yellow glass in it. Your mileage may vary, but that is likely the only place like that in range of teleport (Greater Teleport is another matter as it takes in a LOT more range!). If the wizard can focus on those reference points clearly enough they should be allowed to roll on the ‘very familiar’ row of the familiarity table. Make the Player describe it, or have them roll a concentration check, whatever assuages your disquiet as DM.
If it was a generic hostel, in a featureless slum, then perhaps it could be as bad as “seen casually”.
The ‘similar’ result range seems to support the idea that it is the uniqueness of the destination which matters, if you choose something which isn’t very unique then some kind of arbitrary method of determining which one of the various ‘similar’ destinations you actually meant is made by the magic (aka DM).
‘False destination’ seems a clincher in regards to a teleport destination needing to be unique AND current. If time travel was able to be performed as easily as geographical travel then I suspect if your wizard imagined a place which they were very familiar with but no longer existed there might be a way to get to the last time it DID exist using a Temporal Teleport… but I digress.
What can you do if you can’t find a suitable place that is unique enough already? Make it unique!
That old ruined house of the edge of town away from prying eyes you know of is isolated, but there are a few such ruins on the outskirts of town so it rates as ‘seen casually’. If you’re not going to seal the building to prevent access, then clear some rubble and carve a unique symbol onto the old floorboards then carefully cover it back up with rubble so any casual visitors to the ruin won’t notice it. Unique anchor 1 done.
You want three or more of those unique anchors such that normal events in the area won’t change them. If they do change then you’re in heading into false destination range!
Lastly, there is the negative condition to consider:
“Areas of strong physical or magical energy may make teleportation more hazardous or even impossible.”
Don’t place your destination point near strong magical or physical energies and your mental image of the destination won’t be interfered with when translating into a real destination. It seems a bit like radio static preventing your signal getting through clearly.
Now what “strong physical energy” actually is throws me for a loop, but since wizards don’t usually place a set teleport destination point in the middle of busy market bazaars it’s probably a condition due little concern.
“Strong magical energy” is a more concerning warning as wizards do often ward teleport landing zones. I’ve tended to scale up the meaning of “strong” to compensate for this practice, and consider “strong” to mean something like a place of natural power such as where ley lines cross, a devestating magical battle site, or amidst a bunch of adamantite/mithril veins in the Underdark (or your world’s equivalent if it has one).