This question may sound senseless but I am novice, just looking for clarification. My Cordless phone has a Wall wart AC Power Adapter rated at 9v 500mAh. As far as replacement is concerned, does output power only matters or we have to be specific with ratings also? Like a 4.5V 1A adapter output will also be 4.5W as original rating. I have read in this forum that higher current rating may not matter but higher voltage may damage. Also, I have heard that SMPS based supplies senses the actual load and outputs accordingly. Please correct me.
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Yes both can supply same power but only into the matching load and V may be too low for phone charger. SMPS are regulated and current limited. unregulated wall transformer rectifiers are limited in current but voltage drops to rated value at full current. – Tony Stewart EE75 Sep 18 '17 at 03:24
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To replace your cordless phone adaptor, you need an adaptor rated for 9 volts, and 500 mA (Not mAh!) or more.
The adaptor voltage must match the phone's requirements, but the adaptor's power or current rating may be greater thant the phone's requirements - the phone will only draw the current it requires.
(mAh is a specification for the amount of energy that can be stored ini a battery, and has little relation to the current the battery can deliver. It is meaningless when used for power supplies.)

Peter Bennett
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1... unless you have a power supply that self-destructs after one hour at full load, or three hours at 1/3 of full load, of course. – user253751 Sep 18 '17 at 05:37