If they have no problem discriminating people based on age or medical status, they will most likely discriminate between vaccines.
Of course you’re allowed to discriminate on somethings and not others.
These vaccines have all been approved by official bodies. In fact they have been approved for use by all by official bodies.
Individual countries have taken specific decisions, off the back of that. Some of these decisions are recommendations, not binding (ROI), they’ve also indicated they are temporary. To start faffing around over vaccine types would be utterly ridiculous.