Bridged vs. NAT: a Virtualbox and VMWare Comparison

This all looks normal to me.

Anything under 10.0.0.0/8 (and also 172.16.0.0/12) are perfectly normal NAT addresses. When you put your VMs in NAT mode, the software is essentially acting as it's own dhcp server for the guest machines and will do translations to the host network, so that all the guests on a particular host share an IP with the host. Anything in any of those ranges are fair game for NAT. It appears that VMWare uses a 192.168.0.0/24 range by default, and VirtualBox uses a 10.0.0.0 range. Both are just fine, and neither is better than the other (though I personally prefer 10.0.0.0 ranges because there are 255 times more addresses available).

It sounds like maybe you expected NAT mode to use the NAT between your host network and the internet, but that just doesn't happen. In fact, that is what bridge mode does. Switching to bridged mode means your VM guests are now connected directly to your home router's dhcp server for addresses. VirtualBox and VMWare will both now get addresses from the same place. VMware will have changed as well, it's just that the old mode and the new mode were assigning addresses from similar pools.

The 10.x.x.x range is officially unallocated and intended for internal use just as 192.168.x.x is, so using addresses from this range is no more odd than using a subnet of 192.168.x.x (it just isn't as common - most consumer devices that don NAT and/or DHCP default to a /24 within 192.168.0.0/16).

172.16-32.x.x are reserved for private addressing too, but these are even less commonly used.

The 10.10.2.x range is only visible to virtualbox and the VMs it is running with a NATed network adaptor. for outgoing connections this gets translated, via NAT, to the address used by the NIC of your VMWare host machine, and replies are translated back - this is no different to what VMWare is doing with a 192.168.something.0/24 range.