To escape a basic grapple, you can contest the grappler's Athletics check with an Athletics or Acrobatics ability check (PHB, pg. 195):
Escaping a Grapple. 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 [the grappler's] Strength (Athletics) check.
The mimic's Adhesive trait (MM, pg. 220) says:
Adhesive (Object Form Only). The mimic adheres to anything that touches it. A Huge or smaller creature adhered to the mimic is also grappled by it (escape DC 13). Ability checks made to escape this grapple have disadvantage.
If a player touches a transformed mimic while inspecting it or something, since his hand is already stuck on the mimic, would he be forced to roll an Athletics skill check (regardless of his preferred skill) to escape the grapple?