No effect other than being close.
The space rules are mum about effects of starting a turn in the same square as another creature.
Creatures can't willingly stay in the same space as another.
The creatures in the same space are uncomfortably close for combat.
The moving around other creatures rules for combat state you can't willingly stay in that situation:
Whether a creature is a friend or an enemy, you can't willingly end your move in its space.
Can't choose to eschew moving nor choose to move 0.
In combat, the characters are moving:
In combat, characters and monsters are in constant motion, often using movement and position to gain the upper hand.
The players can choose how much of their move to use, including 0:
You can use as much or as little of your speed as you like on your turn
The rules state you are in motion and you can't end your move still in the same space as another creature. So using your move to travel 0 feet (staying in place) is not an option. Something has to give. Either you have to move the other creature out of the space or move out of the space yourself.
The language used to describe features of tabaxi support moving 0 during a turn:
Once it uses this ability, the tabaxi can’t use it again until it moves 0 feet on one of its turns.
It could be construed that occupying the same square as another in combat involves restricting ones self in a similar fashion to squeezing through a restricted space. If this is reasonable for your table, both you and the other creature would have disadvantage on attack and dexterity rolls until the situation resolves.