Familiars summoned with Find Familiar are not really meant for combat in 5e. The exception is warlock's familiar via Pact of the Chain, but as @spacehamster95 said they still don't level up according to 5e rules. Given their decreased usefulness in scouting and inability to deliver touch spells in BG3 (their main uses in PnP D&D), it's possibly Larian might compensate by upping their HP at higher levels.
BG3 Beast Master Rangers do get a more powerful version of Find Familiar at level 3, and we haven't seen enough levels in BG3 (levels 5+) to know if those companions will level up with us. In 5e their HP does level up: "Its hit point maximum equals its normal maximum or four times your ranger level, whichever is higher." (or HP = 5 + five times your ranger level for beasts of the land via TCoE). Similarly, their attack to-hit and damage increases with your character's proficiency bonus.