Ingesting and injecting is totally different, you are right. For example, potassium chloride is a compound that is used as table salt substitute (though it is often mixed with normal table salt to hide its unnatural bitter flavor and metallic aftertaste), but its oral toxicity is similar to the toxicity of normal table salt, which is low. It is also one of the forms of potassium that are used in potassium supplements. However, potassium chloride is also used in the lethal injection executions: injecting it intravenously causes cardiac arrest (stops heart from beating), which results in death.
Zinc gluconate is the form of zinc used in many dietary supplements and consuming it in amounts not exceeding recommended daily intake cannot in any way cause a result similar to the testicular injection procedure performed in neutering. Zinc is an essential element and a co-factor of hundreds of enzymes. It is reasonable to assume that consumption of zinc gluconate is safe as long as you don't exceed the daily recommended intake, this intake value depends on the size of your dog.
Overdosing soluble zinc compounds, especially on empty stomach, could result in gastrointestinal irritation, nausea, vomiting and diarrhea; it's because zinc ions are corrosive and in excess they irritate gastrointestinal mucous membranes. If concentrated and injected into testicles, their corrosiveness causes fibrous scarring of testicular tissue and kills sperm cells, which is what sterilizes the dog. However, oral consumption couldn't produce these result in any sensible settings. It still could cause poisoning though if overdosed; the most immediate effects would be from gastrointestinal tract (vomiting, etc.), the latent effects would result from the fact that zinc overdose induces deficiency of other micronutrients (such as iron and copper) and suppresses their absorption from diet, which could for example lead to anemia. It is a complicated and delicate balance and it works in inverse as well: for example, overdosing copper in diet could cause zinc deficiency for this reason. My point is that mineral supplements, such as zinc gluconate, are not inherently more dangerous than other microelement supplements; it is the dose that makes the difference between medication and poison.
And referring to their claim of zinc being "neutralized" - they are using imprecise terminology, but I could understand and guess what do they mean, which is that the solution of zinc compound is made to have neutral pH of 7. It is important in oral and dental hygiene to maintain pH above 5.5 because tooth decay starts to occur below pH of 5.5. Zinc gluconate is a salt of somewhat weak acid (gluconic acid) and even weaker base (zinc hydroxide), which makes so that it's acidic (in simplified terms, gluconic acid "wins" and makes the whole compound acidic in solution, which means its water solution has pH below 7). It is therefore mixed with amino acid arginine, which is basic (its water solution has pH above 7) and neutralizes the acidity of this solution.
However, in case of a medication which is not meant to be used in mouth and have contact with teeth, but is instead just consumed to supplement zinc in diet, it doesn't need to be made to have pH above 5.5.
The function of zinc gluconate in oral/dental hygiene gels is to facilitate their antimicrobial action. Zinc ions exhibit oligodynamic effect, which means that even in small concentrations that are safe for humans, dogs, etc. it is extremely toxic to microorganisms and thus kills bacteria responsible for dissolving tooth enamel. Another reason for its presence in these products is that it reacts with volatile compounds responsible for bad breath; the products of this reaction are odorless which temporarily neutralizes bad breath.
In reply to comment asking about the form in which zinc exists in food versus supplements: I don't know exactly in which form zinc occurs in, for example, beef or other foods, but I'd also expect it to be a different chemical compound than zinc gluconate that is used in the dietary supplements, oral cleaning gel, etc. But during the digestive process, both forms are broken down into basic compounds that could be absorbed by our bodies (and dogs' bodies), it is just that zinc gluconate is already a really simple compound. In fact, it is so simple that it could be directly absorbed because it only needs to be dissolved to release free zinc ions; in our food, the zinc is probably incorporated into much more complicated compounds such as large proteins, and these proteins first need to be digested and completely disassembled into their most basic constituents (amino acids) before the zinc ions get released from them to be absorbed by us.