If it's one-time-use, both ABS and PLA are perfectly safe for use as a cookie cutter.
The "food safety" of 3D printed parts is fairly controversial. In fact, whether any particular material is approved by regulators (such as the US FDA) for food contact is much more complex than most people realize. Materials can be accepted for some uses and not others. But there are two major considerations:
Whether leaching is a concern will depend on the kind of contact. For example, nylon will absorb acids but is often used for potable water service. ABS is fine for cold foods but not hot foods. PLA (injection grade) is often used for disposable plastic forks and cold beverage cups, but rarely reusable containers.
There is some anecdotal evidence that lots of 3D printing filaments, particularly cheap Asian filaments, contain toxic chemicals in the pigments and additives. Heavy metals like lead, cadmium, strontium, and all sorts of nasty organic chemicals have been found. You probably WON'T get any meaningful leaching of these chemicals in the brief exposure with cookie dough, but to be safe I would only use reputable US and EU filament manufacturers for food contact. I would also stick to filament colors that use non-toxic pigments (eg plain white is almost always safe titanium dioxide) or no pigments at all ("natural" color).
When it comes to cleaning a cookie cutter for multiple uses, ABS is probably a much better choice than PLA, because ABS can withstand fairly hot water without losing shape. PLA probably won't survive a cycle through the dishwasher. PETG is perhaps a better choice than ABS for longer contact with wet foods, but may or may not survive the dishwasher.
However, the porous structure of 3D printed parts is a serious problem for cleanliness. The tiny grain structure and voids between extruded strands can act as a safe haven for bacteria between uses. This is particularly an issue with cookie dough containing raw egg. A porous surface for raw-food handling would not be acceptable in a commercial kitchen.
Now, the odds of actually transmitting salmonella via poorly-cleaned cookie cutter may be fairly low. Lots of people cut meat at home on wooden cutting boards (which are also porous) and never get sick. Properly cooking the food immediately after handling will go a long way towards reducing the odds of harmful cross-contamination. You'll have to decide for yourself whether the risk profile is worth it. One-time use is the conservative choice.