Just because it connects from the same cellular SIM card(phone number) as your phone, you don't need your phone next to you when you use the cellular, but it would be better if you link the watch to the phone that you use. Apple Watch can operate more functions when it is connected to the phone that set up the watch.
An Apple Watch doesn't need a SIM card plugged inside it, but there's definitely somewhere that the cellular is coming from. You need to at least link the watch's cellular to a phone number(for some cellular carrier companies) to be able to use it on a watch. The other way is using a GPS+Cellular subscription on Apple Watch(but is also just like a cellular plan).
If you skipped the part of setting up cellular, you can still use the watch, but just that you need to keep your phone next to you when you cannot connect to the WiFi.
More information:
https://www.myhealthyapple.com/can-i-use-an-gpscellular-model-apple-watch-without-purchasing-a-cellular-plan/#:~:text=The%20good%20news%20is%20that%2C%20yes%2C%20it%20is,if%20you%20want%20to%20use%20its%20cellular%20features.
Set up cellular on Apple Watch - Apple Support