One of the main differences between GPLv2 and GPLv3 is the so called "anti Tivoization clause", meant to prevent "Tivoization". From https://www.gnu.org/licenses/quick-guide-gplv3.en.html:
Tivoization: Some companies have created various different kinds of devices that run GPLed software, and then rigged the hardware so that they can change the software that's running, but you cannot. If a device can run arbitrary software, it's a general-purpose computer, and its owner should control what it does. When a device thwarts you from doing that, we call that tivoization.
And the relevant part of the GPLv3 license itself, from section 6:
"Installation Information" for a User Product means any methods, procedures, authorization keys, or other information required to install and execute modified versions of a covered work in that User Product from a modified version of its Corresponding Source. The information must suffice to ensure that the continued functioning of the modified object code is in no case prevented or interfered with solely because modification has been made.
If you convey an object code work under this section in, or with, or specifically for use in, a User Product, and the conveying occurs as part of a transaction in which the right of possession and use of the User Product is transferred to the recipient in perpetuity or for a fixed term (regardless of how the transaction is characterized), the Corresponding Source conveyed under this section must be accompanied by the Installation Information. But this requirement does not apply if neither you nor any third party retains the ability to install modified object code on the User Product (for example, the work has been installed in ROM).
So to answer the question: No, the router software being un-modifiable is not a violation of GPLv2 (you should still have access to the source code, of course) while it could be a violation if Busybox and/or the Linux Kernel were GPLv3 licensed. In this hypothetical GPLv3 case the router manufacturer would violate the GPLv3 if they could modify the software of your router while you couldn't.