Basically, the tire has to fit inside the inventory of the vehicle to add the tire onto it's intended slot (such as a tire under the hood of a Lada, or on the back of the V3S)
This was tested in NewUI, unconfirmed of whether old UI also has this bug.
Hello Ozymandias and thank you for the report.
Currently there is no such thing as "spare tire slot", therefore the only way to carry set of spare tires is to put them into the vehicles inventory.
This may be changed in future stages of development.
Regards,
Geez