[RPG] How many skill proficiencies does a College of Lore Bard have

"Having" a skill in 5e is "proficient in" said skill.

Skills are boolean - either you're proficient (and add your Proficiency Bonus and the attribute modifier to the rolls) or not (in which case you still get to add the attribute modifier).

So, since you took the entertainer background, yes, you "have" those two skills, acrobatics and performance.

The Bard class doesn't specify which skills, so any not granted by race or background are fair game to pick. Three more skills, you pick, become proficient.

So you should "have" a total of five proficient skills. The other skills all are non-proficient (at least until you get Jack of All Trades feature).

Bards' Jack of All Trades makes unskilled rolls half-proficient - half your proficiency mod is used unless you're proficient.

And one other thing: If background and class give the same skill, you can pick any other skill instead for the overlap.

Bardic Colleges are not modern Colleges.

A "college education" means something specific in the modern world: it means that someone has gone to a specific kind of educational institution to get a specific kind of education.

A bardic college is not a modern college. Instead, it's basically a club (PHB 54):

Bards form loose associations, which they call colleges, to facilitate their gatherings and preserve their traditions.

Consider the most learned-sounding college, the College of Lore:

The college’s members gather in libraries and sometimes in actual colleges, complete with classrooms and dormitories, to share their lore with one another. They also meet at festivals or affairs of state, where they can expose corruption, unravel lies, and poke fun at self- important figures of authority.

Notice how the first sentence draws a distinction between the bardic college and "actual colleges"?

Your character's progression through levels is how you explain "getting a college education". As they spend time hanging out with people in their college, they learn more and more and gain better and better abilities, which is represented game-wise by your class levels.