D&D 5e doesn't have a specific rule on this because the fifth edition of the D&D ruleset doesn't make use of 5' squares when determining positioning of creatures -- it relies more heavily on an abstraction of what is happening mechanically in the game to determine this. Note that there are variant rules in the DMG for using a grid where 1" square = 5' square.
That being said, what we do have is a general idea of what space a creature of a given size "controls":
Tiny 2½ by 2½ ft.
Small 5 by 5 ft.
Medium 5 by 5 ft.
Large 10 by 10 ft.
Huge 15 by 15 ft.
Gargantuan 20 by 20 ft. or larger
Source: 5e SRD pg 92
And just below that, in the SRD:
A creature’s space is the area in feet that it effectively controls in combat, not an expression of its physical dimensions. A typical Medium creature isn’t 5 feet wide, for example, but it does control a space that wide. If a Medium hobgoblin stands in a 5-foot-wide doorway, other creatures can’t get through unless the hobgoblin lets them.
A creature’s space also reflects the area it needs to fight effectively. For that reason, there’s a limit to the number of creatures that can surround another creature in combat. Assuming Medium combatants, eight creatures can fit in a 5-foot radius around another one.
Because larger creatures take up more space, fewer of them can surround a creature. If five Large creatures crowd around a Medium or smaller one, there’s little room for anyone else. In contrast, as many as twenty Medium creatures can surround a Gargantuan one.
Based on the amount of space a tiny creature takes up (2.5 x 2.5, or 1/4 of a 5 foot square) I think it's safe to say that the breakdown is the same as in previous editions -- you could fit four 2.5' squares inside a single 5' square, so you could fit 4 2.5' tiny creatures in the space occupied by a creature who occupies a single 5' area.
However, I think it's important to remember that things are not often so neatly organized into 5' squares, and that's one of the reasons grid-based combat in 5e is a rule variant and not the base assumption. Allowing combat and positioning to occur more organically might be a bit more work than using a grid system but in my experience it has allowed for much greater creative freedom from players.