Find rows where they are "similar" and in which columns

I'm quite new with SQL Server (2017) and i've this kind of need:
Consider this record:

╔═════════════╦═══════════════╦══════════════╦═══════╗
║   Surname   ║     Name      ║ Day of birth ║ City  ║
╠═════════════╬═══════════════╬══════════════╬═══════╣
║ Rivers Lara ║ Wanda Leticia ║ 07/04/1956   ║ Paris ║
╚═════════════╩═══════════════╩══════════════╩═══════╝

I've to find all the matching records in following list highlighting the type of matching:

╔═════════════╦═══════════════╦══════════════╦════════╗
║   Surname   ║     Name      ║ Day of birth ║  City  ║
╠═════════════╬═══════════════╬══════════════╬════════╣
║ Rivers Lara ║ Wanda Leticia ║ 07/04/1956   ║ London ║
║ Rivers      ║ Leticia       ║ 07/04/1956   ║ Rome   ║
║ Rivers      ║ Leticia       ║ 14/03/1995   ║ Rome   ║
║ Rivers Lara ║ Leticia       ║ 07/04/1956   ║ Paris  ║
║ Rivers Lara ║ Wanda Leticia ║ 08/07/1983   ║ Paris  ║
╚═════════════╩═══════════════╩══════════════╩════════╝

For example:
1st row is matching for Surname+Name+dayofbirth
2nd for Part of Surname+Part of Name+dayofbirth
3rd for Part of Surname+Part of Name
4th for Surname+Part of Name+dayofbirth+City
and so on...

Any ideas on how to approach this type of query will be appreciated considering also that at the moment we have fixed number of possible matching but in the future they could increase (maybe adding more columns like Tax number or other).

Solution

assuming the presentation layer is html and your ok with bits of html in query output, this is a rough idea, though working it's not precisely efficient and no partial matches, only exact. to match partial you'll need to use charindex() or patindex() and split on ' ' with left() or right(), can get convoluted.

one split for left/right word is like, at least this is the way I do splitting still.

--this is only an example on the convoluted nature of string manipulation.
declare @Surname varchar(128) = 'Rivers Lara';
select 
     rtrim(iif(charindex(' ',@Surname) = 0,@Surname,Left(@Surname, charindex(' ',@Surname)))) first_part_Surname
    ,ltrim(reverse(iif(charindex(' ',@Surname) = 0,reverse(@Surname),Left(reverse(@Surname), charindex(' ',reverse(@Surname)))))) last_part_Surname

declare @StartRed varchar(50) = '<span style="color: red;">'
       ,@StopRed varchar(50) = '</span>';
select 
     case when tm.Surname = tr.Surname then @StartRed + tr.Surname + @StopRed else tr.Surname end Surname
    ,case when tm.Name = tr.Name then @StartRed + tr.Name  + @StopRed else tr.Name end [Name]
    ,case when tm.[Day of Birth] = tr.[Day of Birth] then @StartRed + convert(varchar, tr.[Day of Birth], 1) + @StopRed end [Day of Birth]
    ,case when tm.City = tr.City then @StartRed + tr.City + @StopRed else tr.City end City
from TableMatch tm
inner join TableRecords tr on (tm.Surname = tr.Surname or tm.Name = tr.Name) 
    and (tm.[Day of Birth] = tr.[Day of Birth] or tm.City = tr.City)

-- requires either Surname or Name to match and at least 1 of the 2 others to match

additionally, you may be able to use soundex() to find names that sound like other names as a stop-gap without any manipulation. you can also Left() the soundex() value to get broader and broader matches, though you'll end up with all names that start with a the first letter if you goto left(soundex(name),1) matches.