Well, Axor from Hansgrohe is what I call a foo-foo brand, as is the Toto line of faucets. Your Axor lav faucet on line runs $400 to $600 depending on where you buy it. Our concern with these types of brands is not that the quality is bad, but how about support? What is the warranty? Who provides the warranty? If you need a part, how long will it take to get it? I would ask all these questions regarding either brand , of the dealer where you decide to purchase.
Grohe AG is based in Germany, with worldwide plants, including Thailand.
Hansgrohe...I don't know if they are just a spinoff, or just a different company with a coincidentally similar name. Anyway, they are 68% owned by the worldwide conglomerate Masco, and have plants around the globe including Shanghai.
Since I don't personally have much direct experience, except having trouble finding parts for Hansgohe, I will await others with opinions on the brands, etc.
Personally, especially in my own house, I would try to choose a Delta or Moen. If I needed a style not available there, I might look to American Standard, Kohler, or Price Pfister. Others here have differing opinions.