"Tonic" is simply an herb whose essence has been steeped hot water. Long before coffee became the hot beverage of choice, and before tea became a common import from China, Europeans used to drink tonics made from various plants for various purposes.
Bear in mind that those days were before the modern concept of "medicine"--something that you take when you're already sick. In those days, there was more of a concept of doing things proactively. For example, the traditional use for raspberry leaf tonic was for women to drink during pregnancy. But there is nothing considered "medicinal" about it in the sense of their being "something wrong". It won't hurt men or children to drink it, or for that matter women when they aren't pregnant.
Agastaches are mostly from the New World, although Asia has one or two species. A. foeniculum is native to a huge area of the midwestern USA (its relations are more common in the Southwestern USA and Mexico, where there are many different local species), and was almost certainly harvested wild by the Amerinds for medicinal purposes. Unlike most of its dainty relatives, it is a vigorous and almost weedy species, spreading rapidly like mint. Nevertheless, the scent is quite good. The scent reputedly varies by the way. I have 3 varieties; two arrived in a small pot, and one is from seed that was supposed to have been selected for its essential oils. They all smell pretty good.
Some people call it "Anise Hyssop", which is a hideous name for it. It's not a Hyssop, is more attractive than a Hyssop (most of its relatives even more so, being fairly showy of flower), and the scent at least of mine is more of a combination of a fairly clean mild mint smell and fennel than pure anise.
I am not familiar with its traditional uses, so I did an internet search and found this:
An infusion of the leaves is used in the treatment of colds, fevers, weak heart etc. When left to go cold, the infusion is used to treat pains in the chest (such as when the lungs are sore from too much coughing). A poultice of leaves and stems can be used to treat burns.
I guess that is not surprising, as being a member of the Mint family it would tend to have similar uses.
It also attracts bees.
Yes, it does. It's hard to spot but there is one of our native bumblebees on it in the picture.
If I remember correctly, this is one of those medicines that works more as a supplement (like taking excessive vitamin C when you are fighting off a bladder infection) as opposed to a direct help.
It can be drunk anytime; it's pretty mild. The essential oils probably help open up nasal passages which is why it is recommended for colds. They are also probably mildly antiseptic, hence the use as a poultice. I don't know enough about arthritis to conjecture why it might be recommended for that.
Does it kill off all the plants around it like fennel?
I doubt it. Until it really got going in earnest (it's a quick-growing plant), it was getting mobbed by its neighbors, the breadseed poppies. Now they are gone for the season, and it is growing great gangbusters. There is a Sweet Cicely nearby, that doesn't seem to mind it as a neighbor.
