Android - A Real Terminal (Root or No-Root)

^{Summing up from the comments:}

First you need to understand that updating apps works different on Android than it does on a "normal" Linux distro: There is no such thing as "apt". Though Android has its own package manager (listening to the pm command), installs/updates are usually dealt with by services/apps like google-play-store or other alternative-markets. pm would rather be comparable with dpkg on Debian-based systems: it can do local installs (i.e. with the .apk file already present on the device itself: pm install would be the command for that), and a lot of other things.

Apart from that: a terminal app can be compared with a terminal on Linux allright. You get a text-mode "window" where you can execute commands. Most are even similar to those available on Linux, as Android is Linux based: you have things like cd, ls, grep, and more. For details on this, you might wish to take a look at e.g.

• Ten basic Android terminal commands you should know (AndroidCentral)
• Using the terminal (XDA)
• Android Terminal Commands (XDA)
• Android Shell Command Reference (Github)
• our command-line tag-wiki

A look into our terminal tag-wiki will not hurt either :)

I'd say all terminal apps available for Android should support the above. Some even offer additional features, such as an entire development environment (Terminal IDE), or access to remote servers (JuiceSSH).

There is a terminal app which uses "apt" to install packages. Its awesome and has lots of packages ported for arm devices -- Termux & Playstore Link