Most white wines are made to be drunk as young vintages.
Some white wines will age in the bottle and improve in 'tertiary' characteristics - where the wine develops lots of more complex flavours and scents. These tend to be certain types of wine and not necessarily a reflection on their price, though they may well be more expensive wines... white burgundy / reisling / dessert wines such as sauternes / tokaji / etc.
the 'normal' bottles of white wine which tend to be bought and drunk quite rapidly will be your standard chardonnay / sauv. blanc / etc. Generally if you have a white wine which ages well you will know - probably because you will have deliberately bought such a wine - and they can age a long time - I have a bottle of 1906 Tokaji in storage waiting to be drunk at some point - and quite a bit of sauternes which is 15-20 years old now.
Once you know whether the wine is drink soon v. lay down to drink later - storage is quite simple: keep it settled on its side so that the cork remains wet and at temperature as consistent as possible ideally around 12deg. Celsius. Wine will be fine up into the low 20s and down lower as long as above freezing - but prefers 'cellar temperature' which is around 12-15 degrees. Lots of people successfully store wine under the stairs / in a cupboard / in a garage (as long as it doesn't go below freezing) - those who have expensive collections may store them in a specific wine storage fridge (different to a drinks fridge) which keeps it at cellar temperature, and where the compressor is silent and doesn't vibrate - they tend to be c. £1,000+ to buy so you would know if you have one... or they build passive or active cellars to house their wine - passive is where the walls are insulated to get consistent temperatures - active is where there is an air conditioning unit which maintains the temperature in the room...
If you have a normal fridge it will store at about 5 degrees - a drinks fridge at about 8 degrees - both of those are within normal range, so there is no issue in storing white wine (or red) in those temperatures - however, if you can find a space in the house which is under 20 degrees consistently you can as easily keep the wine there and it saves on fridge space!
So having decided the type of wine and how to store it - how do you drink it...
different wines should be served at different temperatures - red a bit warmer than white - though not modern room temperature, but room temperature from the days before central heating! White should be cooler - but not so cold that you can't taste it! Warmer temperatures will open up the wine a bit more and colder will close it down a bit more - you can therefore adjust temperatures to suit if you have a wine that is a bit young / harsh - serve it colder (works for both white and red!) Equally - it is as valid to serve to suit the scenario - sitting outside on a hot day you might want a more chilled white / rose - inside in the winter a warmer red might make more sense - there is no fixed rule... very well explained here: https://www.thewinesociety.com/discover/wine-basics/serve-store-taste/does-serving-temperature-matter
So:
- wine being bought to drink / lay down?
- think about how it is stored
- think about how it is served
standard whites - buy to drink / store at cool room temperature - but no issue if stored in a fridge - just wastes space if you don't need to / serve at fridge to just above temperatures...
other wines you need to take a different approach - note that red wines can be more complex in terms of storage as they are often designed to be layed down to improve and so might have a drinking window when young, then shut down for 5-10 years and then start to re-open, so you have to know your wines - something like a Musar (Lebanese and phenomenal - one of the few affordable top 100 wines in the world) is not even sold until it has been stored for 5-10 years and then can do with 20+ years being layed down - and then when opened could be opened even days in advance of drinking - a whole different world to white wines!
TLDR - no issues storing white wine in a fridge!