When first being grappled it is a contest of the grappler's Athletics vs. the target's Athletics or Acrobatics.
Does the grappled target have the same choice to use their Athletics or Acrobatics when attempting to Escape?
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yes, they can use either Strength (Athletics) or Dexterity (Acrobatics)
PHB, p 195:
A grappled creature can use its action to escape. To do so, it must succeed on a Strength (Athletics) or Dexterity (Acrobatics) check contested by your Strength (Athletics) check.
Just as in their attempt to fight the grapple with either Athletics or Acrobatics, they can choose either in their Escape. The grappler must use their Athletics.