Picking up on the sqlserver.read_only_route_complete Extended Event mentioned by Kin and Remus, it's a nice Debug event, but it doesn't carry a great deal of information with it - just route_port (eg 1433) and route_server_name (eg sqlserver-0.contoso.com) by default. This would also only help determine when a read-only intent connection was successful. There is a read_only_route_fail event but I couldn't get it to fire, maybe if there was a problem with the routing URL, it didn't seem to fire when the secondary instance was unavailable / shutdown as far as I could tell.
I have however had some success joining that up with the sqlserver.login event and causality tracking enabled, along with some actions (like sqlserver.username ) to make it useful.
Steps to Reproduce
Create an Extended Events session to track relevant events, plus useful actions and track causality:
CREATE EVENT SESSION [xe_watchLoginIntent] ON SERVER
ADD EVENT sqlserver.login
( ACTION ( sqlserver.username ) ),
ADD EVENT sqlserver.read_only_route_complete
( ACTION (
sqlserver.client_app_name,
sqlserver.client_connection_id,
sqlserver.client_hostname,
sqlserver.client_pid,
sqlserver.context_info,
sqlserver.database_id,
sqlserver.database_name,
sqlserver.username
) ),
ADD EVENT sqlserver.read_only_route_fail
( ACTION (
sqlserver.client_app_name,
sqlserver.client_connection_id,
sqlserver.client_hostname,
sqlserver.client_pid,
sqlserver.context_info,
sqlserver.database_id,
sqlserver.database_name,
sqlserver.username
) )
ADD TARGET package0.event_file( SET filename = N'xe_watchLoginIntent' )
WITH (
MAX_MEMORY = 4096 KB,
EVENT_RETENTION_MODE = ALLOW_SINGLE_EVENT_LOSS,
MAX_DISPATCH_LATENCY = 30 SECONDS,
MAX_EVENT_SIZE = 0 KB,
MEMORY_PARTITION_MODE = NONE,
TRACK_CAUSALITY = ON, --<-- relate events
STARTUP_STATE = ON --<-- ensure sessions starts after failover
)
Run the XE session (consider sampling as this is a Debug event), and collect some logins:

Note here sqlserver-0 is my readable secondary and sqlserver-1 the primary. Here I'm using the -K switch of sqlcmd to simulate read-only application intent logins and some SQL logins. The readonly event fires on a successful read-only intent login.
On pausing or stopping the session I can query it and attempt to link up the two events, eg:
DROP TABLE IF EXISTS #tmp
SELECT IDENTITY( INT, 1, 1 ) rowId, file_offset, CAST( event_data AS XML ) AS event_data
INTO #tmp
FROM sys.fn_xe_file_target_read_file( 'xe_watchLoginIntent*.xel', NULL, NULL, NULL )
ALTER TABLE #tmp ADD PRIMARY KEY ( rowId );
CREATE PRIMARY XML INDEX _pxmlidx_tmp ON #tmp ( event_data );
-- Pair up the login and read_only_route_complete events via xxx
DROP TABLE IF EXISTS #users
SELECT
rowId,
event_data.value('(event/@timestamp)[1]', 'DATETIME2' ) AS [timestamp],
event_data.value('(event/action[@name="username"]/value/text())[1]', 'VARCHAR(100)' ) AS username,
event_data.value('(event/action[@name="attach_activity_id_xfer"]/value/text())[1]', 'VARCHAR(100)' ) AS attach_activity_id_xfer,
event_data.value('(event/action[@name="attach_activity_id"]/value/text())[1]', 'VARCHAR(100)' ) AS attach_activity_id
INTO #users
FROM #tmp l
WHERE l.event_data.exist('event[@name="login"]') = 1
AND l.event_data.exist('(event/action[@name="username"]/value/text())[. = "SqlUserShouldBeReadOnly"]') = 1
DROP TABLE IF EXISTS #readonly
SELECT *,
event_data.value('(event/@timestamp)[1]', 'DATETIME2' ) AS [timestamp],
event_data.value('(event/data[@name="route_port"]/value/text())[1]', 'INT' ) AS route_port,
event_data.value('(event/data[@name="route_server_name"]/value/text())[1]', 'VARCHAR(100)' ) AS route_server_name,
event_data.value('(event/action[@name="username"]/value/text())[1]', 'VARCHAR(100)' ) AS username,
event_data.value('(event/action[@name="client_app_name"]/value/text())[1]', 'VARCHAR(100)' ) AS client_app_name,
event_data.value('(event/action[@name="attach_activity_id_xfer"]/value/text())[1]', 'VARCHAR(100)' ) AS attach_activity_id_xfer,
event_data.value('(event/action[@name="attach_activity_id"]/value/text())[1]', 'VARCHAR(100)' ) AS attach_activity_id
INTO #readonly
FROM #tmp
WHERE event_data.exist('event[@name="read_only_route_complete"]') = 1
SELECT *
FROM #users u
LEFT JOIN #readonly r ON u.attach_activity_id_xfer = r.attach_activity_id_xfer
SELECT u.username, COUNT(*) AS logins, COUNT( DISTINCT r.rowId ) AS records
FROM #users u
LEFT JOIN #readonly r ON u.attach_activity_id_xfer = r.attach_activity_id_xfer
GROUP BY u.username
The query should show the logins with and without application read-only intent:

read_only_route_complete is a Debug event so use sparingly. Consider sampling for example.
- the two events together with track causality offer the potential to fulfil your requirement - further testing necessary on this simple rig
- I did notice if the database name was not specified in the connection, things didn't seem to work
I tried to get pair_matching target to work but ran out of time. There is some potential for development here, something like:
ALTER EVENT SESSION [xe_watchLoginIntent] ON SERVER
ADD TARGET package0.pair_matching (
SET begin_event = N'sqlserver.login',
begin_matching_actions = N'sqlserver.username',
end_event = N'sqlserver.read_only_route_complete',
end_matching_actions = N'sqlserver.username'
)