It is my understanding the old style trickle chargers if left connected will eventually overcharge your battery. They had no way of monitoring the charge state of your battery. They should be used only to charge a battery overnight but not be left connected for the entire winter.
However trickle charger technology has changed over the years and now some of the newer ones are called "Smart Trickle Chargers" These are more akin to a Battery Tender or Maintainer. These can be left connected all winter.
The Battery Tenders monitor the battery voltage and when fully charged will shut down the charging. As the battery slowly drains off they will kick back in maintaining the charge state of the battery without the possibility of running the battery dry.
I highly recommend a unit similar to the Battery Tender or Battery Tender Jr. I've placed my 04 on one every winter and only because I was taking a trip to Alaska in 2011 did I change the battery. The battery I took out was used as an emergency battery for TLBranth while we were on our way back. I have used it since as the battery in my trailer for charging phones and running lights/laptops etc while camping. It still holds a good charge and I would swap it back into my wing if I had an emergency.
Yes you could hook up two bikes to the same unit but in that case I recommend the larger Battery Tender instead of the Tender JR. The max current is higher in case both batteries need to be refreshed at the same time. Make sure both are already charged before hooking them up.
trickle charger will eventually overcharge a battery. a "maintainer" has circuitry to shut itself off when the battery is fully charged.
it's a bad idea to connect 2 batteries in parallel because unless their fully charged voltage is exactly the same one of them is likely to get overcharged. this is particularly true if the battery type is different such as one old style lead acid and the other a maintenance free.
if you have a lithium ion based battery you need a special type of maintainer for it.
A better option is simply to move the charger from one battery to the other every few weeks. I have 8 bikes I store for the winter and I use 2 maintainers and move them about every few weeks. been doing this for several years and it has worked fine for me.
I have both my bikes on battery tenders when not being used for a while ...as has been said the tender monitors the battery and charges automatically when needed ...the old style trickle chargers do not..I also leave the batteries in the bikes