Is arithmetic with infinite numbers fictitious?
It depends on your definition of "arithmetic with infinite numbers" and "fictitious". The meaning of fictitious that this question was written to oppose, was in reference to certain descriptions of how Robinson's nonstandard analysis is used for calculus. Those descriptions don't have any obvious equivalent for Skolem arithmetic, because Skolem arithmetic is not used as a tool for doing or teaching calculus, or for any other application outside of mathematical logic and model theory.
In 1933 Skolem constructed models for arithmetic containing infinite numbers. In a 1977 article Stillwell emphasized the constructive nature of Skolem's approach. [...]
The words like constructed and construction have no particular meaning here beyond "formal existence proof". Stillwell did not use the word constructive whose precise interpretations do not apply to Skolem's proof.
Is this at odds with Tennenbaum's theorem on nonrecursivity?
There are computable number systems that extend integer arithmetic with additional objects that can be interpreted as infinitely large, and operations extending the familiar ones to the larger system. Polynomials with integer coefficients and computable ordinal notations are two examples. Tennenbaum's theorem shows that Skolem arithmetic cannot be presented in that way, with discrete computable data and operations on them.
This question is related to a comment exchange at Does evaluating hyperreal $f(H)$ boil down to $f(±∞)$ in the standard theory of limits? where terms like "fictitious" are being applied to nonstandard models,
"Fictitious" was applied to descriptions of what is done with nonstandard analysis, not the models themselves. The idea that nonstandard models constructed using the Axiom of Choice have a lesser form of existence than constructs that do not, is certainly an objection that arises in discussions of NSA, just not in the one that you linked to.
The metaphors and fictions relating to NSA occur not (as far as I was asserting) so much in the existence of the objects, but in the descriptions of how the theory is used, such as the idea that there is an ability to take the standard part of bounded $f(H)$ (going beyond the standard rubric of taking limits as $H \to \infty$ when they exist) when this ability never materializes except as the standard thing.
To the extent there is a problem on the existence front, it is that taking individual elements of the nonstandard models is more elusive than just constructing the models, so that the description of "choosing a nonstandard $H$ and calculating $f(H)$ and then taking standard part" can only mean a procedure that is independent of $H$, which is standard analysis dressed in very marginally different words. It doesn't matter whether one considers the individual $H$ to really exist or not, there just is no way to do things like compute standard part of $\sin(H)$ or other functions that depend nontrivially on infinite $H$.
Note 2. The point about a nonstandard model of arithmetic is that one can do a significant fragment of calculus just using the quotient field of such a model.
Only in logic papers. This is not a real "use" of nonstandard arithmetic to do calculus as something taught to and utilized by nonlogicians.