Your application appears to have the same characteristics as mine. I wrote a MySQL custom storage engine to efficiently solve the problem. It is described here
Imagine your data is laid out on disk as an array of 2M fixed length entries (one per entity) each containing 3650 rows (one per day) of 20 bytes (the row for one entity per day).
Your read pattern reads one entity. It is contiguous on disk so it takes 1 seek (about 8mllisecs) and read 3650x20 = about 80K at maybe 100MB/sec ... so it is done in a fraction of a second, easily meeting your 1-query-per-second read pattern.
The update has to write 20 bytes in 2M different places on disk. IN simplest case this would take 2M seeks each of which takes about 8millisecs, so it would take 2M*8ms = 4.5 hours. If you spread the data across 4 “raid0” disks it could take 1.125 hours.
However the places are only 80K apart. In the which means there are 200 such places within a 16MB block (typical disk cache size) so it could operate at anything up to 200 times faster. (1 minute) Reality is somewhere between the two.
My storage engine operates on that kind of philosophy, although it is a little more general purpose than a fixed length array.
You could code exactly what I have described. Putting the code into a MySQL pluggable storage engine means that you can use MySQL to query the data with various report generators etc.
By the way, you could eliminate the date and entity id from the stored row (because they are the array indexes) and may be the unique id – if you don't really need it since (entity id, date) is unique, and store the 2 values as 3-byte int. Then your stored row is 6 bytes, and you have 700 updates per 16M and therefore a faster inserts and a smaller file.
Edit Compare to Flat Files
I notice that comments general favor flat files. Don't forget that directories are just indexes implemented by the file system and they are generally optimized for relatively small numbers of relatively large items. Access to files is generally optimized so that it expects a relatively small number of files to be open, and has a relatively high overhead for open and close, and for each file that is open. All of those "relatively" are relative to the typical use of a database.
Using file system names as an index for a entity-Id which I take to be a non-sparse integer 1 to 2Million is counter-intuitive. In a programming you would use an array, not a hash-table, for example, and you are inevitably going to incur a great deal of overhead for an expensive access path that could simply be an array indeing operation.
Therefore if you use flat files, why not use just one flat file and index it?
Edit on performance
The performance of this application is going to be dominated by disk seek times. The calculations I did above determine the best you can do (although you can make INSERT quicker by slowing down SELECT - you can't make them both better). It doesn't matter whether you use a database, flat-files, or one flat-file, except that you can add more seeks that you don't really need and slow it down further. For example, indexing (whether its the file system index or a database index) causes extra I/Os compared to "an array look up", and these will slow you down.
Edit on benchmark measurements
I have a table that looks very much like yours (or almost exactly like one of your partitions). It was 64K entities not 2M (1/32 of yours), and 2788 'days'. The table was created in the same INSERT order that yours will be, and has the same index (entity_id,day). A SELECT on one entity takes 20.3 seconds to inspect the 2788 days, which is about 130 seeks per second as expected (on 8 millisec average seek time disks). The SELECT time is going to be proportional to the number of days, and not much dependent on the number of entities. (It will be faster on disks with faster seek times. I'm using a pair of SATA2s in RAID0 but that isn't making much difference).
If you re-order the table into entity order
ALTER TABLE x ORDER BY (ENTITY,DAY)
Then the same SELECT takes 198 millisecs (because it is reading the order entity in a single disk access).
However the ALTER TABLE operation took 13.98 DAYS to complete (for 182M rows).
There's a few other things the measurements tell you
1. Your index file is going to be as big as your data file. It is 3GB for this sample table. That means (on my system) all the index at disk speeds not memory speeds.
2.Your INSERT rate will decline logarithmically. The INSERT into the data file is linear but the insert of the key into the index is log. At 180M records I was getting 153 INSERTs per second, which is also very close to the seek rate. It shows that MySQL is updating a leaf index block for almost every INSERT (as you would expect because it is indexed on entity but inserted in day order.). So you are looking at 2M/153 secs= 3.6hrs to do your daily insert of 2M rows. (Divided by whatever effect you can get by partition across systems or disks).