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At certain levels in 5th Edition you can choose to take a boost to ability scores or taking feats. Some classes share feats of their own, most commonly Extra Attack.
The question at hand is; could someone take a feat that's not listed in their class's list of feat? Even if it is a weaker version of it. Ex 1: A Rogue taking Extra Attack. Ex 2: A Fighter taking a weaker version of Sneak Attack. (The Sneak Attack dice wouldn't increase as offer as the Rogue's)
Extra attack and sneak attack are not feats, they're class features. Feats are specifically listed in pages 165-170 of the Player's Handbook.
Some feats do however replicate class features. The Battlemaster's maneuvers can be gotten through the Martial Adept feat and limited spellcasting can be earned through Magic Initiate, for example.
Sneak Attack, Extra Attack and other items on those lists are not feats. Those are called class features. Features are not feats. You gain the class features from your class when you gain the appropriate level.
Feats are detailed in a different chapter of the Player's Handbook. Feats include things like Alert and Polearm Mastery. You can opt to select a feat when your class allows you to.
This is all to say that you are somewhat mistaken: there are no feats on the list of class abilities. To say it another way: you cannot select the abilities from other classes because they are not feats.
Yes and no.
The other answers have already addressed your confusion regarding feats and class features, but I don't think they've addressed the meat of your question.
There is a way to gain another class's class features: Multiclassing. You can find the full rules for this in the Player's Handbook or Player's Basic Rules chapter 6, "Customization Options". In short, it lets you gain a level in a different class instead of your current class. For example, your hypothetical Fighter could choose to take 1 level of Rogue and thus have the same Sneak Attack as a level 1 Rogue. If the fighter was level 5, he'd become a level 6 character with 5 levels in Fighter and 1 level in Rogue.
Multiclassing, like Feats, is an optional rule. Ask your DM if it's allowed. If it is, ask your DM or other players for help when considering multiclassing. Some class combinations are very effective but others can end up being worse than their individual classes. Unlike Feats, multiclassing requires some more thought and careful planning to be effective.
