The font is nice, but can be hard to read, especially in areas where the text is small. The text at the top center really stood out right away as an example of this. Adding an option to switch back to a basic font would be good for those who have trouble reading fancy ones. I'd also recommend adding a small margin to the left side of that top area so the text is not touching the border.
Another thing that can help with readability would be to allow going into fullscreen. As long as you have your Godot project settings set to enable scaling, it can handle the scaling work for you. Then you just need to check the box on Itch for it to add the button to go into fullscreen, without any extra code to handle fullscreen in the game. The project setting is under Display → Window → Stretch. For 2D pixel art games setting Mode to "canvas_items" usually works well. For checking a couple boxes you can get a nice little improvement. I tried using the browser zoom, but the UI elements and cafe scene did not scale. They stayed in their relative positions as the game viewport expanded.
Of course, if you do offer fullscreen, you need to keep in mind that the Esc key exits fullscreen in the browser. When offering browser games with fullscreen, and in general for browser games, it is good to add an alternative pause key, often "P".
I opened the Adopt screen, and again the font really feels harder to read than I think is good. The pink text on white background doesn't help. The larger size helps make the font more readable than if it was smaller, but it feels really big for the space it is in. The single line details are fine, but the "More Info" spanning multiple lines feels like there is so little text on a line that it impacts comprehension. Sure, this is flavor text and not vital details, but it lessens the feel when you have to do extra thinking to process it in an uncommon format. There is a decent amount of space around the whole "Cat Adoption Board" that you can work with, so I would increase its width and give more width to the "Cat Info" area. I'd also suggest adding bolding to the title of each stat/detail in the "Cat Info", or do something else to help the brain quickly process the title separate from the detail it is pointing to. This area could also benefit from more left margin.
For the more info of Bandit, should "Love to explore" be "Loves"?
I adopted a cat and clicked on them to get their comfort up. The final message about customers did not wait for me to do anything and went away before I could read all of it.
I don't yet know how important time passing is. You call it an idle game, so I'm assuming that there is no time pressure, and making the customers and cats wait won't matter. If it does matter, I noticed that opening up the Adopt screen again does not pause things. I don't do many idle games, so the lack of pause had me closing it right away to check on everything, and see if I was missing an interaction I needed to deal with. Playing some more I would say time does not matter, so there is no need to pause when showing the adoption screen. If you do add elements where the player needs to react in a limited time, then you'll want to be sure the adoption screen doesn't cause them to miss it.
I do appreciate that you included a pause screen, as that allows me to not miss things while I am typing these notes.
I adopted a second cat. A new customer showed up as soon as I closed the screen. I also saw the original customer get stuck for a short time on one of the cats. They sorted it out, so it is not a big deal, but as the number of characters on screen grows, you may need to work on the navigation/path finding to reduce the chances of this.
I played a bit more and then started adoption more cats. I adopted 5, and had 3 customers when 2 customers decided to go to the counter at the same time, and got stuck on each other. Then the third customer joined in. One of the cats got stuck on the pileup for a while, but freed themselves.
I'm still making money, and there don't seem to be any significant negatives at the moment, other than not being able to interact with the customers. I went ahead and adopted the remaining cats, and got the You Win message. The game was continuing in the background. It would be nice if you added an "OK" or "Keep playing with the cats" button, for those who want to just enjoy having all the cats.
I tried clicking on the Upgrade and Cafes buttons, but they did not seem to do anything. If they are not yet implemented, I would suggest disabling them so they don't look active to players. You could also have them show a message about what they will be and that they are not ready yet, but I think the simple disable conveys that well enough.