Is ethernet describing the protocol and therefore cat5 cables became labelled ethernet cables because they are used for this protocol?
The IEEE 802.3 standards define certain L1 characteristics/operation, but do not define the connectors to use to connect the devices.
For instance, while 802.3 does define that 100BASE-TX will use twisted pair cabling that must meet certain characteristices (attenuation, cross talk, etc), it does not define if this must be UTP or some variation of STP. It also doesn't define a certain "Category" of cabling, although typically the cabling "Categories" are designed to meet specific requirements of the 802.3 standards.
With reference to the standard not defining the connector, this is probably best illustrated using fiber as an example. The 802.3 standards define the characteristics/operation of 1000BASE-SX, but you can find 1000BASE-SX connections that use LC, SC, MTRJ, ST or a number of other types of connectors. As long as the connector allows the connection to meet the requirements of the standard, it is perfectly okay to use it.
As to why the cables are called Ethernet cables, I have already addressed that in this answer to a previous question, so it is probably better to view that answer as well rather than for me to copy it here.
Ethernet runs on various different media, of which UTP cabling is only one medium. There are other LAN and other technologies, such as POTS, which can use UTP cabling.
UTP cabling has various registered categories. Category-5 cabling is no longer registered, and it hasn't been this century. The current registered categories are Category-3, Category-5E, Category-6, and Category-6A. Different variants of ethernet on UTP require a minimum category. For instance, 10 Mb ethernet can run on Category-3 or above, but 100 Mb ethernet requires at least Category-5E.
'Ethernet Cables' really describe RJ45. RJ45 has standards associated to it to carry a Layer 2 (see OSI model) signal called Ethernet.
Other Layer 2 signals include
MOE (metro optical ethernet)
RJ45 has two twisted pair that must have a certain amount of twists in them per foot to prevent signal crossover from power cords, radio signals, etc. Additionally, 'the signal' can only traverse a certain amount of distance before the signal attenuates, which is in the standard.
Ethernet is a family of specifications that governs a few different things