Yes the WeatherBridge Nano is the same thing as the Meteobridge Nano, with one small difference....
If you buy the WeatherBridge Nano from Ambient then you get their special version that looks identical, runs exactly the same firmware, but has a license for AmbientWeather.net. Then you can besides uploading weather data to numerous places also send your data to AmbientWeather.net. One of the neat advantages of sending your data to AmbientWeather.net is that they've built in the API and skills for it to integrate with Amazon Echo's Alexa and Google Assistant call Google Home (Hey Google). Then you can talk to your smart home device and get all your weather data from your weather station by just talking.
The Meteobridge has been around for many years and is well supported. It has grown very capable. It is yet very easy to use. The added features it offers over the WiFi Logger do not complicate the user interface. The Meteobridge just does a lot more.
The WiFi Logger although less than a year old is also well regarded. If they both had started at the same time its possible that they would have the same feature sets. So in a few years who knows what the WiFi Logger will be capable of...or maybe not...I think the Meteobridge has more powerful CPU and innards (I haven't really compared them in this regard but that is what I've read people say without confirming).
You should do your research. Read the long WiFi Logger thread in this forum to see what it might be like to have one of these.
https://www.wxforum.net/index.php?topic=34312.0Then you can also read up in tons of places about the Meteobridge Nano. Realize that the Meteobridge comes in many forms. The Nano is the newest version that only works for Davis consoles. Then there is the original Meteobridge and the Meteobridge Pro which are external devices. The Pro has antenna and build in radio to pickup directly from Davis ISS but the regular Meteobridge needs to plug into the Davis console via USB Logger or via the Network via IP Logger. There are pros and cons of each depending on what you are trying to accomplish.