In 5e I know you add your ability modifier and proficiency bonus to your skills that you chose when creating your character, but what if you don't have that skill, do you still add the corresponding modifier ie; acrobatics=Dex, or no bonus at all?
There are a few classes you can multiclass into to gain skills, such as Bard, Ranger or Rogue.
In addition to that, in the player's handbook there is a feat called "Skilled" which allows you to gain proficiency in any combination of 3 skills or tools.
In Xanathar's Guide to Everything, we have the Prodigy feat which gives a skill proficiency and a few other things, but can only be taken by a Human, Half-Orc, or Half-Elf.
In Tasha's Cauldron of Everything, the feat Skill Expert gives proficiency in a skill and a few other benefits, with no racial prerequisite.
There is also a bard ability in the college of lore that allows you to learn 3 more skills at level 3.
Currently, the downtime rules allow you to gain proficiency in tools but not skills.
In case it isn't clear, you don't have to pick skills from your class skill list when you gain new skills through the "Skilled" feat. You can pick any skill you want.
A race defined by class levels with a racial bonus in a skill does not automatically treat it as a class skill. The rules text quoted in the question has been taken out of context and does not apply to PC character creation.
The text referenced in the Monster Manual is written in the context of building and reading monster statblocks. The Skills heading of Reading the Monster Entries not only clarifies this, but also says that having class levels exempts a monster from receiving these free class skills.
This line gives the creature’s skills, along with each skill’s modifier (including adjustments for ability scores, armor check penalties, and any bonuses from feats or racial traits). All listed skills are class skills, unless the creature has a character class (noted in the entry). A creature’s type and Intelligence score determine the number of skill points it has. (7)
The text quoted on page 307 is a further elaboration on the above text from page 7 meant to explain how to interpret and rebuild a monster's statistics. It mentions skill ranks and racial bonuses because a skill is usually only listed in a monster's entry if that monster has a rank or a racial bonus to that skill. Indeed, the Glossary entry on class skills goes on to explain when you should interpret a skill listing as a class skill or whether the skill is listed simply because it's relevant to the monster's special abilities. This text has nothing to do with PC character creation, and should not be taken out of context to apply to how to build PC characters.
Not only has the text been taken out of context, but also you can easily see plenty of evidence that proves contrary to what this question suggests. For example, there exist numerous examples of NPCs that do not receive free class skills just for having a race with bonuses on skill checks. Some PC races explicitly give class skills as a racial trait.