So I think there's two problems that have to be answered in your question: whether or not climbing can invoke an ability check, and whether or not climbing should require your action.
Whether or not a particular character behavior requires an ability check is dependent on the DM, unless the rules have reified that behavior as requiring an ability check. So it's fully within the DM's purview to say "climbing this sheer cliff requires a Strength check, you can add your athletics proficiency", or to elide that check for any reason at their discretion.
However, climbing a cliff is governed within the general rules on movement:
Each foot of movement costs 1 extra foot (2 extra feet in difficult terrain) when you're climbing, swimming, or crawling. You ignore this extra cost if you have a climbing speed and use it to climb, or a swimming speed and use it to swim. At the DM's option, climbing a slippery vertical surface or one with few handholds requires a successful Strength (Athletics) check. Similarly, gaining any distance in rough water might require a successful Strength (Athletics) check.
―Climbing, Swimming, and Crawling, Player's Handbook, pg 182
As you can see from the quoted rules here, there's already a cost baked into the act of climbing, whether or not it includes making an ability check to successfully do it. Although the DM is technically allowed to impose Action Economy costs on top of this (as they are with any other behavior not explicitly reified as such in the book), they really shouldn't have in this case. Climbing is a type of movement, and should be treated as such.
My best guess, without personally knowing your DM, is that they may have been thinking of rules from other editions of D&D, or possibly other d20-based games (like Pathfinder) where such checks may have required the use of the character's action. My guess is based on the colloquial use of the term "skill check", which don't exist in 5e anymore―5e has "ability checks", which may or may not add your skill proficiencies to the roll depending on the type of check or the DM's discrepancy. Before your next session, I would talk to your DM about revising this case (and other cases that involve special movement speeds).
One other thing I'm going to add, because I saw it mentioned in comments below and figure it's good advice that should receive visibility: if the DM is going to associate an Action Economy cost to behavior that isn't reified as such in the books, they should convey that to the player before the player makes the decision to do it. I can understand unusual circumstances where an action to climb over a wall might be appropriate (perhaps the DM is inferring the player is taking the Dash action to get some extra distance?) but if that's the case, the DM should be stating that outright so that the player can make choices about how they want to spend their turn.