Completeness in mathematics is essentially a metric concept (that means that every Cauchy sequence in the metric space converges to an element of the space).

Sometimes (but I think more on a physical standpoint, and I agree is a sort of repetition and not so common) it is used to characterize bases in vector spaces, in the sense that a basis is complete if its linear span is the whole vector space (the orthonormal basis being a special case where the vector space is given an additional inner product structure). Be warned that a (complete) orthonormal basis in a vector space with inner product $V$ may not be a (complete) basis of the metric (Hilbert) space obtained as the completion of $V$ with respect to the metric induced by the inner product.

Apart from logic, that is probably quite far from what it is intended here, I cannot think of other instances of the word complete in math (but maybe I am forgetting something).

However, since in one case "complete" is associated to a (metric) space, and in the other to a basis in a vector space, I think it reasonably easy to avoid confusion; even if the latter can be indeed a complete space itself in some metric.

Completeness in mathematics is essentially a metric concept (that means that every Cauchy sequence in the metric space converges to an element of the space).

Sometimes (but I think more on a physical standpoint, and I agree is a sort of repetition and not so common) it is used to characterize bases in vector spaces, in the sense that a basis is complete if its linear span is the whole vector space. The axiom of choice implies that every vector space has a basis.

The orthonormal basis on a Hilbert space is actually different: the space has such additional structure that we can afford to do infinite linear combinations of vectors, provided they converge in norm. However, be warned that a (complete) basis of orthogonal vectors in a vector space with inner product $V$ (in the finite linear combination sense), it is not in general a (complete) basis of the metric (Hilbert) space obtained as the completion of $V$ with respect to the metric induced by the inner product. However it may happen that it is a basis in the Hilbert space sense, i.e. if infinite linear combinations are allowed.

Apart from that and mathematical logic, that is probably quite far from what it is intended here, I cannot think of other instances of the word complete in math (but maybe I am forgetting something).

However, since in one case "complete" is associated to a (metric) space, and in the other to bases in vector spaces (possibly with additional care to specify if we allow finite or infinite combinations), I think it reasonably easy to avoid confusion.