The Quasit Does Not Gain Blindsight
You have already quoted the relevant text:
Its statistics are the same in each form, except for the speed changes noted.
Knowing the definition of Statistics makes this clear. From page 6 of the Monster Manual:
A monster's statistics, sometimes referred to as its stat block, provide the essential information you need to run the monster
In other words, according to the wording of the ability, nothing in the Quasit's stat block changes, except for the speed changes noted. The Quasit does not become a bat, it merely changes shape to look (and locomote) like one. The only change of note, aside from the speed changes, is that the Claw action is re-flavored as a Bite to better fit the new form. The Quasit still has full use of its Invisibility, and retains its ability scores, resistances, immunities, and all other features that make up the Quasit stat block. Refer to this excellent answer by David Coffron (or pages 6-11 of the Monster Manual) for a full list of things that are considered a monster's Statistics.