You're missing the fact that many DHCP servers, without further configuration, will give each device, as identified through its MAC address, the same IP address – even when they ask for a new lease.
Of course you can configure DHCP servers differently, but it's not uncommon for these leases to be reserved for a certain period of time, say a week, or a month. This is called reservation, and it may be enabled by default. It's just really useful: Imagine you take your laptop to work every day and then come back home. You will get the same IP address and won't have to look it up every time.
If you're talking about your router and your LAN IPs, look into its settings whether there's a table for IP address allocation that's automatically filled, and clear it. If you're talking about your external IP given to you by your ISP, there's not much you can change here, really. The ISP uses the MAC address of your modem (which they handed to you) to know it's you who connects. Changing the MAC address won't help you get a new IP—you just won't be able to connect to the Internet.
As @tictacbum correctly mentions,
ipconfig will only show your LAN IP address, not the WAN address (the external IP). To check this, use any online service for that, or just Google.