The requirements for SOC 2 are defined in the Trust Services Criteria for Security, Availability, Processing Integrity, Confidentiality, and Privacy (pdf as of May 2022). This document is a set of very high level requirements that are satisfied by an organization implementing controls. It's an accurate statement that there is room for interpretation - since they apply to many organizations in many different fields, there is some need for organizations to be able to determine the best way to comply with the intent of each requirement. At the end of the day, the controls (processes, procedures, tools, tool configurations, etc.) that the organization puts into place are what get audited and assessed.
Controls around what software is approved for installation on a company-issued device or a device that connects to the company network or stores company (or client) data are probably designed to satisfy requirements around protecting information assets from security events (CC6.1), protecting against threats from outside the system boundaries (CC6.6), restricting moving and removing information to authorized parties (CC6.7), preventing and detecting unauthorized and/or malicious software from entering the system (CC6.8).
The characteristics around requirement CC6.8 in the Trust Services Criteria do specifically call out restriction of software installation, detecting unauthorized changes to system configurations, and having a defined change control process. If I had to pick a specific requirement that was being addressed, this would probably be the one.
There's nothing in SOC2 that says that certain people can't be approved to install software on their own computers. In fact, I've worked for organizations that did allow software developers to have limited administration rights on their company-issued devices for the purposes of installing software development tools. However, there were specific controls in place to limit the people with administrator rights to people who needed to regularly install non-standard software packages, monitor what was being installed, ensuring that installed software was up-to-date, and so on. However, the controls all documented this. Your organization's controls are probably written in a way that doesn't carve out giving software developers the access needed to install software or provide the ability to ensure that the software they are installing is safe and supports ongoing business operations.
It may be a bit annoying, but it's also a bit unusual if someone were to discover on a Friday afternoon that they are missing a key piece of software needed to carry out their work. When I've worked for organizations that do limit administrative access, they tend to compensate by making sure that a wide variety of software development tools are available in a trusted repository and available for easy (perhaps even self-service) installation and regularly refresh this list. Otherwise, you would typically have at least a couple of days lead time between realizing that you need a tool (or a new version of tool) and actually being in a position where you must use it.