This is an attempt to elaborate on my comment as requested.
"Inversion" refers to an operation that maps each point A to the point B exactly opposite A with respect to the inversion centre, and at the same distance from that centre as A. In coordinates, with the inversion centre at the origin, inversion can be described by changing the sign of all coordinates.
The winning strategy for the first player in the penny game is to place a coin in the centre and from then on place a coin such that it is related to the second player's previous coin by inversion. Two facts are required to make this work:
a) After each move by the first player, the situation must exhibit inversion symmetry; that is, it must be invariant under inversion; that is, if you apply the inversion operation to the entire situation, you end up with exactly the same situation. That guarantees that wherever the second player finds a free spot to play, the opposite spot is also free for you to play.
b) Equally importantly, the second player must not be able to place a coin such that it covers two points related by inversion; that is, it must be impossible to place a coin such that it covers points $A$ and $B$ where the inversion operation maps $A$ to $B$. If such a move were possible, then you couldn't replicate that move in the opposite spot, since the move would occupy $A$ and $B$ at the same time, and you'd also have to occupy $A$ and $B$ to maintain the inversion symmetry, but you're not allowed to do that.
Your modification of the problem by modifying the sizes of the coins obviously doesn't prevent you from maintaining property a) -- you can go on placing coins in the opposite spots like before.
To see that it also doesn't change property b), note that the inversion centre always lies on the line connecting any two points $A$ and $B$ related by inversion. A shape is by definition convex if it contains all points on all line segments between any two of its points. (Clearly a coin is convex in this sense.) But since you placed a coin on the inversion centre in the first move, the second player can never use a convex object to simultaneously cover two points $A$ and $B$ related by inversion, since that would necessarily also cover the inversion centre, which is already occupied.
Thus, as long as there is at least one object exhibiting inversion symmetry that you can use for your first move, and all remaining objects are convex and available in pairs (or in infinite supply), the winning strategy can still be applied.