According to the comments and my own experience with Docker, each RUN line is run in a separate shell environment, so when you source an environment in one RUN line, that environment is not available to other RUN commands.

Using the line RUN source scl_source enable devtoolset-7 && cd /home/admin/${APP_NAME}/nginx-base/cplusplus && make version && make instead of your previous RUN command makes sure the current environment is setup for the make command.

Add the scl repo for centos:

sudo yum install yum-utils # if needed
sudo yum-config-manager --add-repo https://vault.centos.org/7.9.2009/sclo/x86_64/rh/repodata/repomd.xml

Install this libgfortran dependency required for devtoolset-8:(source)

sudo yum install -y wget
wget https://vault.centos.org/7.9.2009/os/x86_64/Packages/libgfortran5-8.3.1-2.1.1.el7.x86_64.rpm
sudo yum install libgfortran5-8.3.1-2.1.1.el7.x86_64.rpm -y

Install devtoolset-8 with nogpgcheck:(it would be much better to install the key for the repo)

sudo yum install -y devtoolset-8 --nogpgcheck

Finally enable the scl for devtoolset-8:

scl enable devtoolset-8 bash

Running this command shows both the gcc 7 and gcc 8 in the path:

which -a gcc

Output from which -a gcc

/opt/rh/devtoolset-8/root/usr/bin/gcc
/usr/bin/gcc

devtoolset is called gcc-toolset in RHEL8.

The following commands worked for me:

microdnf install -y gcc-toolset-12
scl enable gcc-toolset-12 bash
gcc --version
# gcc (GCC) 12.1.1 20220628 (Red Hat 12.1.1-3)

To expand on @user2915097's answer here is a working example using devtoolset-7 and rh-python36 instead of devtoolset-1.1

FROM centos:7

# Default version of GCC and Python
RUN gcc --version && python --version

# Install some developer style software collections with intent to
# use newer version of GCC and Python than the OS provided
RUN yum install -y centos-release-scl && yum install -y devtoolset-7 rh-python36

# Yum installed packages but the default OS-provided version is still used.
RUN gcc --version && python --version

# Okay, change our shell to specifically use our software collections.
# (default was SHELL [ "/bin/sh", "-c" ])
# https://docs.docker.com/engine/reference/builder/#shell
#
# See also `scl` man page for enabling multiple packages if desired:
# https://linux.die.net/man/1/scl
SHELL [ "/usr/bin/scl", "enable", "devtoolset-7", "rh-python36" ]

# Switching to a different shell has brought the new versions into scope.
RUN gcc --version && python --version

FROM centos:centos7
RUN yum update -y
RUN yum groupinstall "Development Tools" -y    
RUN yum install wget -y
RUN curl -O https://ftp.gnu.org/gnu/gcc/gcc-7.3.0/gcc-7.3.0.tar.gz
RUN tar xzf gcc-7.3.0.tar.gz
RUN cd gcc-7.3.0
RUN ./contrib/download_prerequisites
RUN cd ..
RUN mkdir gcc-build
RUN cd gcc-build
RUN ../gcc-7.3.0/configure                           \
    --enable-shared                                  \
    --enable-threads=posix                           \
    --enable-__cxa_atexit                            \
    --enable-clocale=gnu                             \
    --disable-multilib                               \
    --enable-languages=all
RUN make
# (Go make a cup of ice tea :)
RUN make install

To save the build time you can create a new docker from the running docker using "docker commit" or save /usr/local to a tar file and open it on any other fresh centos7 docker.