No idea if the taste is the same, but my first two all grain brews had what I recognised as 'home brew' taste from my kit days. I changed two things - I acquired a fridge and temperature controller to keep my fermentation temperature locked at 18C in the first 3-4 days (when it is most exothermic) and I did simple water treatment - pre-boiling with a Campden tablet and adding a bit of Gypsum and Epsom salts. The taste disappeared and has never come back. Unfortunately, I don't know which of those interventions was the answer! A less lazy person would do a series of test brews, leaving one of them out each time .... the only experiment I have made is dosing the water with CRS rather than pre-boiling in order to reduce it's native carbonate - that seemed to taste fine too (still used the Campden tablet).
At least these are cheap and easy things to try and also to continue with in the longer term (apart from the fermentation fridge, I suppose, where you may have constraints of space if not budget).
At least these are cheap and easy things to try and also to continue with in the longer term (apart from the fermentation fridge, I suppose, where you may have constraints of space if not budget).