No, you do not have to choose a god or goddess. There are plenty of atheists and non-religious folk who practice different aspects of magic, spell casting, and even witchcraft.
Yes, you can choose a god or goddess to connect and work with; however, there are many people on here who also believe that a deity chooses you. Some people do choose their deities based on those who seem to have similar personalities and what that deity may represent. Choosing a god and goddess may very well just be for a given situation or magical working but not necessarily on a level where you work with them daily and have a personal relationship with.
To me, I do think that the deities whom you work with and incorporate in your practices can be from different religions as long as you do not destroy the religious canon (the structure of the religion) of those deities from different myths and religions. It is also might be good to learn the difference from UPG and PCPG as well.
Invoking is simply to call upon a deity rather it be through prayer or whatever. People invoke deities to help them in their workings as a means to help carry out the working, for protection, and that kind of thing.
Here are some resources that may help assist you further:
UPG and PCPG in Practice by Personified - http://www.spellsofmagic.com/read_post.html?post=598601
Choosing a Goddess & a God by Flora Sage (charmingpixieflora) - https://www.youtube.com/watch?v=4VhpJ3imua4
Finding Your God and Goddess Deity by The Chira Project -https://www.youtube.com/watch?v=l9xDGGjV-BU