Bubbles leave when added to syrup

It'll be because the tonic syrup contains nucleation sites, which are in this case tiny particles from the stuff in the syrup that cause the dissolved gas to come out of solution rapidly. The more nucleation sites, the quicker this will happen. Disturbances in the liquid can also act as nucleation sites - which is why your carbonated drinks foam up when poured, and why stirring/shaking the liquid results in extra bubbles for a short time.

Edited to add: to get rid of the nucleation sites you could try filtering the syrup (I note that this is part of the recipe). This can be as simple as several layers of paper-towels arranged into the shape of a cup and sat inside a funnel, or you could use more expensive pre-made filters(e.g. coffee filters) or scientific grade filter papers/cartridges. If I were you I would use several layers at one time of whatever filter you choose, this will help filter out as much as possible of the detritus, but will slow your filtration rate - be patient.

WARNING: DO NOT TRY THIS AT HOME; SIGNIFICANT RISK OF BURNS. If you have no nucleation sites you can retain gasses in liquid quite well. As evidenced by all those people who have heated water in a microwave and then added sugar or something similar (coffee, cocoa) and had it rapidly (and sometimes explosively) boil over. Also known by all those people who have dumped sugar into their soda/pop/fizzy/carbonated drink and seen the massive efflux of bubbles.

I don't know the exact chemical/physical reason - but it definitely happens, with any carbonated liquid.

Presumably it's an increase in the surface area of 'places bubbles can form' which tends to be why a fizzy drink produces bubbles at the edges, against the glass it's in, around a wedge of lime, etc.
Try dropping a teaspoon of sugar in a glass of coke, or the fun one when I was a kid; stir it with a really well-chewed [real, wood] liquorice stick. Make sure you have a towel to hand. The coke will go almost totally flat inside 30 seconds.
If 'daddy owns the vineyard' then observe the same reaction pouring champagne into a dry glass.

The trick for Sodastream is to squeeze a couple more pushes of gas into the bottle first. It's not truly successful, because a home device like a Sodastream has a lower pressure limit than the commercial bottling plants do.
If you try this with a Sodastream bottle, by adding the syrup to make an actual 'bottle of tonic' be really careful pouring the syrup into the bottle; a bit at a time… again, you're either careful enough or you need a towel.

Note; you cannot re-fizz Sodastream once you have added anything other than water to the bottle. You may get away with it a few times, but eventually you will foul the mechanism.

If you're working from store-bought soda water, there's nothing you can do really.