If I build a Bard he can't learn any new spells outside his class. The wizard however can learn any spell he gets in his/her spell book. If the Bard takes the feat Magic Initiate and chooses wizard, then would he have a spellbook that he could keep adding spells to?
Magic Initiate does not provide a spell book. If it did, it would say so.
If you are interested in having a bard that knows a greater variety of spells, consider the feat Ritual Caster, which does provide a ritual book, and lets you add ritual spells to it that you may find during your adventures.
Be aware you can only cast these spells as rituals, however. They would be a separate batch of spells from your Bard spells.
A spell book is a unique feature of the Wizard class in 5e. Be aware though, it’s not correct that a wizard can learn any spell. They can (generally) only add spells from the wizard spell list to their books.
Also note, a Bard’s 10th level power, Magical Secrets, will grant you 2 spells that do not need to be on the Bard spell list.
Tip about the different spellcasters
The rules for spellcasting for different classes can be confusing at first. There is no table, for instance, where you can compare what classes get what features. When you read the rules, understand the class gets, or the feat provides, exactly what the rules say, nothing more. It might feel like it would be sensible if a wizardly magic initiate got a spellbook, but the rules don't say that, and you should not assume it's implied.