5e rules are not written like a legal document. It's meant to just be everyday language and common sense. Some parts are even left undefined on purpose for the GM to decide it, in which case there is no RAW answer aside from the RAW answer being "ask your GM."
You get a spellbook at level 1 because it's part of your equipment. You can choose not to take that equipment; in that case, you don't have the spellbook. Since it's from your given class equipment, that means you don't get it when you multiclass, just like when you multiclass into other classes and don't get that related equipment.
https://www.sageadvice.eu/2015/01/20...book-for-free/
That's my take on it anyway.
Are you thinking of multiclassing with a wizard in AL? How does that stuff work anyway in Adventure League normally? If there's a vague area, does the GM still get to decide?