Assuming that the desired sequence is traffic from your LAN should go from the local machine -> tun0 -> tun1, this is likely what is happening, however it is happening in a way which is not visible on a tracreroute.
Lets take a packet destined to an arbitrary internet address - I'll use 184.108.40.206 in this example.
The computer picks up the packet and looks for how to send it. It sees that it should be sent via tun1 (because the 2 routes below are the equivalent of a default route, but more limited, so preferred over the default route - in this case the first route is hit) -
0.0.0.0/1 via 10.8.1.1 dev tun1
220.127.116.11/1 via 10.8.1.1 dev tun1
But here is the part which may not be obvious. If you look at the configuration for tun1, you will find it has an endpoint which is 18.104.22.168. There is a specific route for this IP address which goes over tun0
22.214.171.124 via 10.8.3.1 dev tun0
Similarly, there is another route
126.96.36.199 via 188.8.131.52 dev eth0 proto static metric 100
This route makes traffic sent via tun0 directly accessible through the ethernet interface.
As this is a very specific route, traffic to 184.108.40.206 will go through tun0. Thus all traffic flowing to the internet (over tun1) must go through 220.127.116.11 which is itself a tunnel, so yes, data will be flowing through both tunnels.
A traceroute won't show this because the packet does not know it is being tunnelled through a tunnel. That said, you can still check that this is happening by looking at the lower levels - Generating traffic while in another window doing a "sudo tcpdump -n -i any". You will see that whenever a packet is sent to the wider internet, a packet will be sent through each of eth0, tun0, tun1, and the same will be true of returned packets. The packets associated with tun0 will all have the target of 18.104.22.168.