"I am wondering whether using the Adobe programs with this account, paid for by my employer, somehow means they [the employers] have some form of intellectual property claim/rights on my personal work because it was made with this license?"
I am neither an attorney nor giving you free legal advice. But depending on who you work for and the terms of your employment contract, it could be more far-reaching than that.
To give a real example, when I worked at a major university, I had to sign a contract giving them half interest in anything I created while I was employed there. It had nothing to do with whose equipment/software I used either. I could have written a cookbook at home during off-work hours with paper and pens I purchased myself, it didn't matter. Technically, they owned half of it.
Ask your human resources dept for copies of everything you have signed. If you are unsure about anything, ask for clarification and/or consult a legal expert.