I purchased lodgings above the Library then I purchased another one above the gambling den. My profile already shows my lodgings as the gambling den, however, when I Travel to my lodgings, it still takes me to the library. I nearly dropped my pipe in my anxiety!
No matter which lodgings you’re currently living in, the ‘Lodgings’ part of the map will always look identical. The only major effects moving to one home or another has is A: how many cards you can have in your hand at once (1 card for the starter lodgings, up to 5 for the most expensive ones) and B: whether you draw the cards that are part of the ‘City Vices’ set (a few of them, obtained through seasonal events or Fate-locked stories, prevent these cards from appearing).
Each lodging you own unlocks a card that you might draw. There’s an option on the card to rent out, shutter, or otherwise stop using the lodging, which stops you from drawing its card. You can’t do that on the one where you currently live.