The usual construction of the hyperreals replaces real numbers with sequences of real numbers, and also introduces a nontrivial equivalence relation on the sequences, making two sequences equivalent if they agree on a “large” set of terms. The real numbers get represented by the constant sequences, infinitesimals get represented by sequences that approach 0, and infinite numbers are represented by sequences that grow without bound.
The magic is in how “large set of terms” is defined. You need a “large set” relation with the property that finite sets are not large, and for any set either the set or its complement is large. Then we can resolve your question: say you had two not-always-zero sequences that multiply to give the all-zero sequence. Then the set of zero positions is large for one of those two sequences. And that means one of your sequences is equivalent to the zero sequence. The field axioms are saved!
This honors project has what looks like an accurate write up of the construction along with proofs of some of the main theorems: https://ideaexchange.uakron.edu/cgi/viewcontent.cgi?article=...
A simpler example of another real division algebra (and in fact, another field) is the field of rational functions with real coefficients. This field is also infinite dimensional over the reals (for example, 1, x, x², x³... are linearly independent).