Would it make sense to say "Я очень экономичный муж", or "У меня такая экономичная собака" when someone wishes to allude to his being frugal with apparel, meal, diversions etc. Or a dog is small and consumes little. And all this allows to spare the family money on the whole.
No, you can't say экономичный meaning "frugal".
Экономичный means "cost effective", "frugal" would be экономный.
The latter is a neutral term, the former is objectification (the figure of speech opposite to personification).
That said, you can certainly say у меня очень экономичный муж or у меня очень экономичная собака meaning "I have a very cost effective husband" or "I have a very cost effective dog".
This would mean that they perform at the same level as conventional husbands and dogs, but require less investment and less maintenance. They consume less resources and yield higher ROI.
They might not look as fancy as other husbands and dogs, and be less comfortable, but they produce more measurable output for one ruble invested in them.
If you want to say that your husband is frugal and controls his expenses (rather than consuming less resources which you, in your generosity, allot for him), you should use the word экономный
Well, it looks like the word you are looking for is "экономный", like in phrase "Мой муж очень экономный и старается не делать лишних трат". It's an exercise left to the reader imagine saying the same about dog - but "экономная собака" is as valid as, don't know, "щепетильная собака" - why the hell not after all.
With inanimate object adjective "экономичный" is used (in a meaning close to ergonomic, like in "экономичный двигатель".
Both of these word should not be confused with "экономический" - using them interchangeably is a mistake.
If you mean frugal (person) or that a dog doesn't require a lot of money to keep, 'экономичный' makes no sense.
Экономичный implies equipment, method of production, etc. which saves energy, time and resources.
You can say 'экономный муж' , meaning he doesn't buy expensive things and always thinks how to spend money, i.e. FRUGAL.
As for the dog, it's tricky. I can't think of an adjective to express the idea of 'dog inexpensive to have'. A description may be required, e.g. на такую собаку не надо тратить много денег и времени (apparently some dogs are cheaper to have).
It should be noted that Russian and English adjectives with the 'economy' root may and do have different shades of meaning.