No for example Swisscom in Switzerland is using Ericsson for their 5G network and Huawei for their IP core transport network. A lot of households got upgraded to Huawei G.Fast technology for example.

So it's also about price. Huawei is just cheaper than Nokia and Ericsson. I don't know how the technology of the different providers compare but it probably won't be that much of a difference. It probably also depends on your existing 4G infrastructure since 5G can sometimes be rolled out as a software upgrade using existing antennas. So if your existing infrastructure uses Huawei you don't want to switch.