If the laptop uses a 2.5" hard drive, then you can easily remove the hard drive and use a USB to SATA Adapter, drive dock, or enclosure to connect the drive to another Mac.
If you have multiple user accounts on either Mac or if you have added & deleted user accounts over the years, then it may be a little more difficult accessing the data in that user account(s). macOS uses a "UserID" which is a number to identify users on a system (the user name is just for our benefit). If the "userID" of an account on your newer Mac matches the "userID" of the account on the old Mac, then you can just use the Finder to navigate to your old user folder and just copy items like normal. If the "userIDs" are different, then you can still copy the locked folder (if you want everything inside it) by normal methods since macOS will prompt you for your admin password. Another option is to use an app like Carbon Copy Cloner (CCC) which will automatically prompt you for an admin password to access the locked folders plus CCC will allow you to select and deselect items to copy.
You could also just change permissions on that old user folder as well to match your current user. This is not a good idea if you ever want to use this drive in the old laptop again.
You can only use the old hard drive to boot a newer Mac if the OS on the old drive is one which is supported on the newer Mac. The oldest OS a Mac can boot from is the version which originally shipped with the Mac from the factory (although there are a few exceptions when a Mac model is not updated when a new OS is released).
You should always have frequent and regular backups. This is even more important with the newer Macs using SSDs and especially with the USB-C Macs where it may not be possible to recover any data from a failed SSD or a non-booting USB-C Mac.
Edit: Is the charger or charger cable bad or is it the Magsafe port on the laptop? Most likely the laptop is still functional. On older Macs the Magsafe port is inexpensive and can be replaced. Maybe you can take the laptop to an Apple Store to have them see if the laptop will work with another charger before you purchase a new charger.
