Yes, it is possible. In fact, Earth Engine collections (such as the Sentinel 1 'COPERNICUS/S1_GRD') are stacks of multitemporal images with associated metadata that allows the user to filter them to extract relevant information:
////////////////////////////////////
// Load the Sentinel-1 ImageCollection
var s1 = ee.ImageCollection('COPERNICUS/S1_GRD')
.filterBounds(geometry) // Draw your AOI in the map area
.filterDate('2014-10-03', '2020-06-21');
////////////////////////////////////
In the previous example the Sentinel 1 collection is imported and filtered spatially and temporally. In this case the date filter is provided as an example since these dates include the whole collection.
However, Earth Engine will not be able to directly use a collection to perform a classification. Before you can proceed you will have to create a multiband image. I provide an example below:
////////////////////////////////////
// Filter to get images from different look angles
var asc = s1.filter(ee.Filter.eq('orbitProperties_pass', 'ASCENDING'));
var desc = s1.filter(ee.Filter.eq('orbitProperties_pass', 'DESCENDING'));
var vvvhAsc = asc
// Filter to get images with VV and VH single polarization
// Filter to get images collected in interferometric wide swath mode.
.filter(ee.Filter.eq('instrumentMode', 'IW'));
// Create a composite from means at different polarizations and look angles.
var composite = ee.Image.cat([
vvvhAsc.select('VV').median(),
vvvhAsc.select('VH').median(),
vvvhDesc.select('VV').median(),
vvvhDesc.select('VH').median(),
]).clip(geometry);
// Rename the bands so you can identify them when they are all joined together
var s1comp = composite.select(
['VV','VH','VV_1','VH_1'], // old names
['s1vva','s1vha','s1vvd','s1vhd'] // new names
);
////////////////////////////////////
In this case I have created a 4-band image, which incorporates both look angles and VV and VH polarisations. In this way you can make sure you have as much information as possible for your classification. Each band is the result of reducing the more than 5 years of data to the median value of each pixel, which is very effective way to eliminate radar speckle.
Depending on what it is that you want to classify, for example surface water presence, you might want to use temporal information (wet months) instead of look angles and polarisation. In this case you will have to filter the collection to extract the periods you are interested in (using ee.Filter.dayOfYear(start, end)) to create your monthly composite. In this case, you can use the following example instead of the previous block of code:
////////////////////////////////////
// Filter them by two-moths periods and extract the average values
var s1_JanFeb = s1.filter(ee.Filter.dayOfYear(1,60))
.median();
var s1_MarApr = s1.filter(ee.Filter.dayOfYear(61,120))
.median();
var s1_MayJun = s1.filter(ee.Filter.dayOfYear(121,180))
.median();
var s1_JulAug = s1.filter(ee.Filter.dayOfYear(181,240))
.median();
var s1_SepOct = s1.filter(ee.Filter.dayOfYear(241,300))
.median();
var s1_NovDec = s1.filter(ee.Filter.dayOfYear(301,360))
.median();
// Create a multiband composite image
var composite = ee.Image([s1_JanFeb, s1_MarApr, s1_MayJun, s1_JulAug, s1_SepOct, s1_NovDec])
.clip(geometry);
////////////////////////////////////
You have a full example of multitemporal composite creation with code included here:Article Large-Scale, Multi-Temporal Remote Sensing of Palaeo-River N...
The resulting composite can be used to perform supervised classifications, which are a relatively straight procedure in Google Earth Engine.
Hopefully I have provided a satisfactory answer to yor question. Let me know if this is not the case!