A creature with blindsight can perceive its surroundings without relying on sight, within a specific radius. Creatures without eyes, such as oozes, and creatures with echolocation or heightened senses, such as bats and true dragons, have this sense. (5e SRD, pg 86)
It is important to note that being naturally blind is not a required condition for a creature to have Blindsight or for Blindsight to apply. In cases where creatures are blind outside of their Blindsight radius, it is called out in the entry, such as in this example from the Black Pudding monster:
Senses: blindsight 60 ft. (blind beyond this radius)
As written, Bats can be blinded because they are not immune to the condition. However, Blindsight, being a more specific rule, still allows a bat to function while blinded and will not cause the bat to suffer from disadvantage on attacks or grant advantage to attacks against it.
In short, Blindsight is a specific exception to the blinded condition, and differs from being immune to blinded.
However, bear in mind that Blindsight has a radius. For this example I'll use a hypothetical creature that has a ranged attack and Blindsight. The blinded Blindsighted creature may make attacks without disadvantage against targets within its Blindsight radius, but outside of that radius its attacks may suffer from the penalties. The same goes for attacks against it: attacks made by attackers outside the Blindsight range may get advantage, while inside the radius they do not.