Avoid making local installs into system directories. The system directories eg /usr, are reserved for the package management system to use. By definition, if you are doing make install that means you are making a local install, and if you need to do sudo make install that means you don't have permission to wherever you are writing.

So, if you are getting permission errors with make install, check and see whether you are trying to install into system directories, and install into /usr/local or similar instead. /usr/local is reserved for local installations. You may need to give yourself permission to write to /usr/local, but this is usually easily done. On Debian this can be done by adding yourself to the staff group. Better still, find or create a binary package, and install that instead. That way you can easily keep track of installed packages and obtain the other benefits of package management.

Note that the package management system conversely does not install into /usr/local, per the FHS. See Section 9.1 of the Debian Policy Manual- File system hierarchy for an overview.

The error messages show missing make gcc etc. Install the build-essential meta-package to get the development tools.

sudo apt-get install build-essential

Make is a general purpose workflow program, usually used for compilation. But it can be used for anything.

When you do something like "make all", the make program executes a rule named "all" from a file in current directory named "Makefile". This rule usually calls the compiler to compile some source code into binaries.

When you do "make install", the make program takes the binaries from the previous step and copies them into some appropriate locations so that they can be accessed. Unlike on Windows, installation just requires copying some libraries and executables and there is no registry requirement as such. In short, "make install" just copies compiled files into appropriate locations.

When you call make with arguments, you are giving it a list of targets.

There is some discussion of this in the Upgrading Ports section of the FreeBSD Handbook where it says:

Unlike the usual make install clean command, it will upgrade all
the dependencies prior to building and installing the selected
port.

[...]

Note: You can save two extra steps by just running make install
clean instead of make, make install and make clean as three
separate steps.

Also, as a side note: Don't build your package as root unless you really need to. In general you should work in an unprivileged account and then as a final step you can do sudo make install if you have to. It would be better, though, to add the unprivileged user to a group with write-access to the install directories, or even better to install it into a non-system area (for example, with ./configure you can use --prefix=) since most systems, whether Linux or BSD, usually have packages and a packaging system that is used to install software.

In your case you don't need to do a sudo make install as you will install the library in your $HOME. You have configured the build to use a prefix and bindir to install it in your $HOME.

To make the package avaiable to other programs it would be better to choose directories outside of your $HOME. Then you will need to do the make install using sudo.

TLDR: altinstall skips creating the python link and the manual pages links, install will hide the system binaries and manual pages.

Let's take a look at the generated Makefile!

First, the install target:

install:         altinstall bininstall maninstall

It does everything altinstall does, along with bininstall and maninstall

Here's bininstall; it just creates the python and other symbolic links.

# Install the interpreter by creating a symlink chain:
#  $(PYTHON) -> python2 -> python$(VERSION))
# Also create equivalent chains for other installed files
bininstall:     altbininstall
        -if test -f $(DESTDIR)$(BINDIR)/$(PYTHON) -o -h $(DESTDIR)$(BINDIR)/$(PYTHON); \
        then rm -f $(DESTDIR)$(BINDIR)/$(PYTHON); \
        else true; \
        fi
        (cd $(DESTDIR)$(BINDIR); $(LN) -s python2$(EXE) $(PYTHON))
        -rm -f $(DESTDIR)$(BINDIR)/python2$(EXE)
        (cd $(DESTDIR)$(BINDIR); $(LN) -s python$(VERSION)$(EXE) python2$(EXE))
        ... (More links created)

And here's maninstall, it just creates "unversioned" links to the Python manual pages.

# Install the unversioned manual pages
maninstall:     altmaninstall
        -rm -f $(DESTDIR)$(MANDIR)/man1/python2.1
        (cd $(DESTDIR)$(MANDIR)/man1; $(LN) -s python$(VERSION).1 python2.1)
        -rm -f $(DESTDIR)$(MANDIR)/man1/python.1
        (cd $(DESTDIR)$(MANDIR)/man1; $(LN) -s python2.1 python.1)