How to access files of my First Ubuntu from my Second Ubuntu

If you can determine the partition number using lsblk you could mount the drive and directly access it there. Example:

$ lsblk
NAME        MAJ:MIN RM   SIZE RO TYPE MOUNTPOINT
sda         259:0    0 476.9G  0 disk 
├─sda1      259:1    0   512M  0 part /boot/efi
└─sda2      259:2    0 476.4G  0 part
└─sda3      259:3    0 476.4G  0 part /
$ udisksctl mount -b /dev/sda2

In this (hypothetical) example, sda2 is a partition that currently is not mounted. It could thus be your system partition of the previous Ubuntu installation. Partitions on internal drives are not automounted, unless they are explicitly configured to be so. This will mount your drive in a folder under /media/<your user name>, so navigating there and clicking the folder where the partition is there will show its contents.

You can also use the the "traditional" mount command, but then you need to be root:

$ sudo mount /dev/sda2 /mnt

This mounts the partition under the /mnt folder.