I'm answering this one myself because since the day I posted this question I found different ways to make the MKR1010 reachable through it's name in the local network (i.e. Arduino.local).
So, first of all make sure that your sketch contains:
- the capacity to connect to an existing WiFi network
- a properly working web server instance
- a properly working webpage to handle the http calls
Then, I found that according to what your real needs are, the following library and methods might help you, each one in its own way:
WiFiNINA, gives you the possibility to assign a fixed IP address through the WiFi.config() method. This is very cool, because you don't need to manually assign it from the router. However, I understand it will not check if the same IP already exists in the same local network, and - more important - it will not work if you don't align to the DHCP capacities of the router. In fact, most common routers assign local IP in the below range:
192.168.x.x
172.16.x.x. - 172.31.x.x
10.0.0.0 - 10.255.255.255
An iPhone DHCP is able to provide only 16 addresses in the 172.20.10.0 to 172.20.10.15 range.
Example: if you want to assign an IP address such as WiFi.config(192,168,10,10) but the router assigns IP in the 172.16.x.x range... it will not work. In this case, it will work if you will WiFi.config(172,16,10,10).
So, lesson learned, know what is the range the router will assign local IP, and align to it staying within that range. It will work!
ArduinoOTA gives you the possibility to actually name the dns of your web server. This way any device who wants to connect from the local network can resolve the ip to reach it from its name, if they have the ability to solve names (more on that, later). Super cool, and easy.
When you initialise it :
ArduinoOTA.begin(WiFi.localIP(), "my-arduino", "password", InternalStorage);
the "my-arduino" string can be changed to the name you like. Then you can reach the web server typing in the browser my-arduino.local
ArduinoOTA is fully compatible with WiFiNINA library in all my tests.
mDNS Actually this library will not work with WiFiNINA officially, as it is designed for Ethernet shields. However, it does something that is very unique and that I really needed: it resolves the dns name of other devices or web-servers connected on the same local network (for example other Arduinos or RaspPi web servers. The WiFiNINA library, for example, does not have the ability to resolve names on the local network, only this in the www. However I managed to modify the example sketchs to resolve the IP of locally connected devices with the MKR1010 and WiFiNINA library. I can't now remember everything I changed in the example sketchs but... trust me: if I managed... then anyone can!

That's it, hope it'll give paths to follow for others in the future.