What would be disrespectful about calling her Brenda for example ?

The potential for confusion when documenting the work in any scientific papers.

Many research animals are named. Alex the parrot comes to mind.

This is an outcome of Jane Goodall challenging the convention of the day to use code numbers for animal subjects. At the time it was considered apostasy.

How does that have amything to do with disrespect?

It doesn't. The spider's name is 'Number 16'. It could be any alphanumeric string.

But the original assertion was that the researchers "behaved respectfully" by using the #16 name.

Spiders are math loving creatures

The respect is of other researchers who have to consume the data at a later date. "Number 16" is far more valuable a label than "Brenda", since the former implies the specimen is one of at least 16 other data points - whereas the latter implies that the researcher involved is simply bored of their work and just wants to have fun.

Anyway, only humans use names. Is "Brenda" a human or a spider? Sure, you can answer this now - but can you answer it in another 10 years?

Humans use names for non human things as well and using a number doesn't imply spider anymore than anything else. Is #16 a spider or a sheep? Did anyone ever confuse Dolly for a human?

It also doesn't tell us that it's one of at least 16 data points. From the article we only know that it was the 16th spider they recorded as being present but we don't know that any of the other 15 made it into the study. The fact that you're reading something we don't know into it from the name #16 actually highlights a failing in this system for me. Surely if it matters to you then you'll be reading the study which will these details explicitly.

Not having respect for the scientific process, where using loaded or ambiguous labels introduces culture where only raw technical facts are required.

Meh. As another commenter pointed out, https://en.wikipedia.org/wiki/Alex_(parrot) . Also I find the idea of "academic seriousness" very weird. I would never ever want to wake up with the thought of doing "serious science" all day. I want science (in my personal case, math) to be interesting, and thus fun.

"Man is most nearly himself when he achieves the seriousness of a child at play." - Heraclitus

Thats fine, just keep your personal preferences out of my labs' white paper submissions.

"Number 16" is valuable, in that it communicates the fact that the specimen involved is one of a set of at least 16 individual data points. "Brenda" doesn't communicate anything much more than, perhaps, the researcher is bored with their work and wants to have fun instead of doing real science. Besides which, this is a human name, so is it referring to a human or a spider?

Scientific rigour is important. Fanciful notions of cultural joviality, less so.

It'd be unduly familiar.