Arch is minimal in the sense that the default system consists of very little. If you install nothing but the base metapackage, you won't get a usable system out of it at all. The purpose of Arch is to fill out a complete, working system by making your own choices. If you follow the installation guide, then you start out with base, linux, and linux-firmware. Base is a metapackages that consists of archlinux-keyring, bash, bzip2, coreutils, file, filesystem, findutils, gawk, gcc-libs, gettext, glibc, grep, gzip, iproute2, iputils, licenses, pacman, pciutils, procps-ng, psmisc, sed, shadow, systemd, systemd-sysvcompat, tar, util-linux, xz.
To get a fully-working system, you'd also need a bootloader, but Arch doesn't prescribe what that has to be. Alpine is mostly going to give you these same thing in terms of available CLI utilities, but rather than being based on GNU libc and GNU coreutils, it's based on musl and busybox. The init system is OpenRC rather than systemd. And it has a default bootloader, which is syslinux.
This makes Alpine more "minimal" in the sense of a minimal installation taking up less disk space, because musl, busybox, and OpenRC are smaller in the literal sense of the binary files consume less disk space than glibc, GNU coreutils, and systemd. Busybox also comes with ash (I think actually dash) as the default shell, which smaller than bash.
I have no idea if the apk package manager is smaller than pacman. They're both smaller than what you'd get out of a Debian or Redhat descended system.
Personally, I think it's a bit misleading to call either of these more minimal than the other. The functionality, feature set, list of available utilities is pretty much the same. Alpine is just giving you smaller files, though note that using a musl-based system presents a lot of difficulty because a fair amount of software Linux users expect and are familiar with isn't really POSIX-compliant and only works with GNU C.